diff --git a/.gitignore b/.gitignore
index 2368d6d..7695f4a 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,8 +1,9 @@
 *~
 *.o
-/gss
+/kk
 /main
 /testsurfaces
-/serial.cmd*
+/serial.cmd.[eo]*
+/parallel.cmd.[eo]*
 /save
 */save
diff --git a/Makefile b/Makefile
index f207be9..9eb97de 100644
--- a/Makefile
+++ b/Makefile
@@ -1,24 +1,28 @@
 
-BISON = /opt/local/bin/bison
-FLEX = /opt/local/bin/flex
+devel = 0
 
-CXX = g++
+BISON = bison
+FLEX = flex
 
-INCLUDES = -I/opt/local/include -I.
+# CXX = g++
+# CXX = OMPI_CXX=clang++ mpic++ -fno-color-diagnostics --stdlib=libc++ --std=c++11 -I/u/cseed/llvm-3.1/lib/c++/v1
+CXX = clang++ -fno-color-diagnostics --stdlib=libc++ --std=c++11
+
+INCLUDES = -I. -I/opt/local/include 
 
 # OPTFLAGS = -g
 OPTFLAGS = -O2 -g
 # OPTFLAGS = -O2 -DNDEBUG
 
 LDFLAGS = -L/opt/local/lib
-# LDFLAGS = -pg -L/opt/local/lib
 
-CXXFLAGS = $(OPTFLAGS) -Wall -Wno-unused $(INCLUDES)
+CXXFLAGS = $(OPTFLAGS) -DHOME="\"`pwd`\"" -Wall -Wno-unused $(INCLUDES)
 
 LIB_OBJS = lib/refcount.o \
   lib/lib.o lib/smallbitset.o lib/bitset.o lib/setcommon.o lib/io.o lib/directed_multigraph.o
 ALGEBRA_OBJS = algebra/algebra.o algebra/grading.o algebra/polynomial.o
-KNOTKIT_OBJS = planar_diagram.o dt_code.o knot_diagram.o cube.o spanning_tree_complex.o \
+KNOTKIT_OBJS = planar_diagram.o dt_code.o knot_diagram.o cube.o steenrod_square.o \
+  spanning_tree_complex.o \
   smoothing.o cobordism.o knot_tables.o sseq.o \
   knot_parser/knot_parser.o knot_parser/knot_scanner.o \
   rd_parser/rd_parser.o rd_parser/rd_scanner.o
@@ -26,7 +30,8 @@ KNOTKIT_OBJS = planar_diagram.o dt_code.o knot_diagram.o cube.o spanning_tree_co
 COMMON_OBJS = $(KNOTKIT_OBJS) $(ALGEBRA_OBJS) $(LIB_OBJS) 
 
 LIB_HEADERS = lib/lib.h lib/show.h lib/refcount.h lib/pair.h lib/maybe.h lib/vector.h \
-  lib/set.h lib/ullmanset.h lib/bitset.h lib/smallbitset.h lib/setcommon.h \
+  lib/set_wrapper.h lib/set.h lib/hashset.h \
+  lib/ullmanset.h lib/bitset.h lib/smallbitset.h lib/setcommon.h \
   lib/map_wrapper.h lib/map.h lib/hashmap.h lib/ullmanmap.h lib/mapcommon.h \
   lib/unionfind.h lib/priority_queue.h lib/io.h \
   lib/directed_multigraph.h
@@ -36,11 +41,12 @@ ALGEBRA_HEADERS = algebra/algebra.h algebra/grading.h algebra/module.h \
   algebra/polynomial.h algebra/multivariate_polynomial.h \
   algebra/multivariate_laurentpoly.h algebra/fraction_field.h
 KNOTKIT_HEADERS = knotkit.h planar_diagram.h dt_code.h knot_diagram.h \
-  smoothing.h cobordism.h cube.h spanning_tree_complex.h cube_impl.h sseq.h
+  smoothing.h cobordism.h cube.h steenrod_square.h \
+  spanning_tree_complex.h cube_impl.h sseq.h simplify_chain_complex.h
 
-LIBS = -lgmp
+LIBS = -lgmp -lz
 
-all: gss
+all: kk
 
 %.o : %.cc
 	$(CXX) -c $(CXXFLAGS) $< -o $@
@@ -48,15 +54,19 @@ all: gss
 %.o : %.cpp
 	$(CXX) -c $(CXXFLAGS) $< -o $@
 
-gss: gss.o $(COMMON_OBJS)
-	$(CXX) $(LDFLAGS) -o gss $^ $(LIBS)
+kk: kk.o $(COMMON_OBJS)
+	$(CXX) $(LDFLAGS) -o kk $^ $(LIBS)
 
 main: main.o $(COMMON_OBJS)
 	$(CXX) $(LDFLAGS) -o main $^ $(LIBS)
 
+mpimain: mpimain.o mpi_aux.o $(COMMON_OBJS)
+	$(CXX) $(LDFLAGS) -o mpimain $^ $(LIBS)
+
 testlib: testlib.o $(COMMON_OBJS)
 	$(CXX) $(LDFLAGS) -o testlib $^
 
+ifeq ($(devel),1)
 knot_parser/knot_parser.cc knot_parser/knot_parser.hh: knot_parser/knot_parser.yy
 	$(BISON) knot_parser/knot_parser.yy -o knot_parser/knot_parser.cc
 
@@ -74,6 +84,7 @@ rd_parser/rd_parser.cc rd_parser/rd_parser.hh: rd_parser/rd_parser.yy
 
 rd_parser/rd_scanner.cc: rd_parser/rd_scanner.ll
 	$(FLEX) -o rd_parser/rd_scanner.cc rd_parser/rd_scanner.ll
+endif
 
 .PHONY: parser_files
 parser_files: \
@@ -87,7 +98,7 @@ parser_files: \
 .PHONY: clean
 clean:
 	rm -f *.o lib/*.o algebra/*.o knot_parser/*.o rd_parser/*.o
-	rm -f main gss
+	rm -f main kk mpimain
 	rm -f gmon.out
 
 .PHONY: realclean
@@ -104,4 +115,6 @@ realclean: clean
 
 $(LIB_OBJS): $(LIB_HEADERS)
 $(ALGEBRA_OBJS): $(ALGEBRA_HEADERS) $(LIB_HEADERS)
-$(KNOTKIT_OBJS) main.o gss.o: $(KNOTKIT_HEADERS) $(ALGEBRA_HEADERS) $(LIB_HEADERS)
+$(KNOTKIT_OBJS) main.o mpimain.o kk.o: $(KNOTKIT_HEADERS) $(ALGEBRA_HEADERS) $(LIB_HEADERS)
+
+mpimain.o mpi_aux.o: mpi_aux.h
diff --git a/README b/README
index 8986544..4da80c4 100644
--- a/README
+++ b/README
@@ -1 +1,162 @@
-I modified cotton's readme.
\ No newline at end of file
+knotkit is a C++ software package written by Cotton Seed
+(cseed@math.princeton.edu) for computing some knot and manifold
+invariants appearing in low-dimensional topology.  Other contributors
+include Josh Batson.
+
+TABLE OF CONTENTS
+
+1. INTRO
+2. BUILDING
+3. USAGE
+4. UPCOMING CHANGES
+5. FOR DEVELOPERS
+
+1. INTRO
+
+In addition to accepting knot presentations in a variety of formats
+(see usage below), knotkit contains the following tables of knot data:
+
+* The Rolfsen knot tables through 10 crossings, extracted from
+  Bar-Natan's Mathematica package, KnotTheory`,
+* The Hoste-Weeks-Thistlewaite knot tables through 16 crossings taken
+  from knotscape, and
+* and the Morwen Thistlewaite hyperbolic link tables taken from
+  SnapPy.
+
+This version of knotkit has support to compute the following
+invariants:
+
+* Khovanov homology
+See:
+M. Khovanov, A Categorification of the Jones Polynomial, Duke
+  Math. J. 101 (2000), 359--426, arXiv:math/9908171.
+D. Bar-Natan, On Khovanov's Categorification of the Jones Polynomial,
+  Algebraic and Geometric Topology 2 (2002), 337--370.
+
+* Szabo's geometric spectral sequence 
+Computing gss was the original motivation for writing knotkit.  For
+more information on gss, see:
+Z. Szabo, A geometric spectral sequence in Khovanov homology, 
+  arXiv:1010.4252, and
+C. Seed, Computations of Szabo's Geometric Spectral Sequence in Khovanov
+  Homology, arXiv:1110.0735.
+
+* The Batson-Seed link splitting spectral sequence
+See:
+J. Batson, C. Seed, A Link Splitting Spectral Sequence in Khovanov
+  Homology.
+
+* The Lipshitz-Sarkar Steenrod square on Khovanov homology
+See:
+R. Lipshitz, S. Sarkar, A Khovanov Homotopy Type, arXiv:1112.3932.
+R. Lipshitz, S. Sarkar, A Steenrod Square on Khovanov Homology,
+  arXiv:1204.5776.
+
+* Bar-Natan's analogue of Lee homology
+See:
+E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197
+  (2005), 2, 554–586, arXiv:math/0210213.
+D. Bar-Natan, Khovanov’s homology for tangles and cobordisms,
+  Geom. Topol. 9 (2005), 1443–1499, arXiv:math/0410495.
+R. Lipshitz, S. Sarkar, A refinement of Rasmussen's s-invariant,
+  arXiv:1206.3532.
+
+* The s-invariant coming from Bar-Natan's analogue of Lee homology
+In addition to the above references, see:
+J. Rasmussen, Khovanov homology and the slice genus, Invent.
+  Math. 182 2 (2010), 419--447, arXiv:math/0402131.
+R. Lipshitz, S. Sarkar, A refinement of Rasmussen's s-invariant,
+  arXiv:1206.3532.
+
+2. BUILDING
+
+To build knotkit on OS X, you will need the latest version of XCode
+(available for free in the App Store) and the GNU multiple precision
+arithmetic library (GMP).  You can get GMP here:
+
+http://gmplib.org/
+
+knotkit also builds under Linux.  In addition to GMP, you will need a
+C++ compiler which supports C++11.  I use LLVM clang, but knotkit
+should build with a recent version of GCC.
+
+knotkit doesn't have a sophisticated build system.  To build knotkit,
+just run:
+
+make
+
+from the knotkit source directory.  This should build the executable
+"kk".  For instructions on invoking kk, see usage below.  If you run
+into a problem, please contact me: Cotton Seed
+(cotton@math.princeton.edu).
+
+3. USAGE
+
+The usage message for kk is given below.  This can also be obtained by
+running "kk -h".
+
+A note about output.  The output for commands kh, gss, ls and lee is a
+.tex file which renders the bigraded homology group or spectral
+sequence.  The output for the command sq2 matches the output for the
+program written by Lipshitz-Sarkar and is suitable for loading into
+Sage.  The command s outputs a single line of text.
+
+usage: kk <invariant> [options...] <link>
+  compute <invariant> for knot or link <link>
+<invariant> can be one of:
+  kh: Khovanov homology
+  gss: Szabo's geometric spectral sequence
+  lsss: Batson-Seed link splitting spectral sequence
+    component weights are 0, 1, ..., m
+  sq2: Lipshitz-Sarkar Steenrod square on Z/2 Kh
+    output suitable for Sage
+  leess: spectral sequence coming from Bar-Natan analogue of Lee's
+    deformation of Khovanov's complex (whew!)
+  s: Rasmussen's s-invariant coming from lee
+output:
+    kh, gss, lsss, leess: .tex file
+    sq2: text in Sage format
+    s: text
+options:
+  -r         : compute reduced theory
+  -h         : print this message
+  -o <file>  : write output to <file>
+                (stdout is the default)
+  -f <field> : ground field (if applicable)
+                (Z2 is the default)
+  -v         : verbose: report progress as the computation proceeds
+<field> can be one of:
+  Z2, Z3, Q
+<link> can be one of:
+  - the unknot, e.g. U or unknot
+  - a torus knot, e.g. T(2,3)
+  - a Rolfsen table knot, e.g. 10_124
+  - a Hoste-Thistlethwaite-Weeks knot, e.g. 11a12 or 12n214
+  - a Morwen Thistlethwaite link, e.g. L8n9 or L13n8862
+  - a planar diagram, e.g.
+      PD[X[1, 4, 2, 5], X[3, 6, 4, 1], X[5, 2, 6, 3]] or
+      PD[[1, 4, 2, 5], [3, 6, 4, 1], [5, 2, 6, 3]]
+  - a Dowker-Thistlethwaite code, e.g.
+      DTCode[6,8,2,4],
+      DT[dadbcda] or
+      DT[{6, -8}, {-10, 12, -14, 2, -4}]
+  - a braid, e.g. BR[2, {-1, -1, -1}]
+  - disjoint union (juxtaposition), e.g. T(2,3) U
+
+4. UPCOMING CHANGES
+
+The following changes are currently planned:
+
+* support for Z/p, p arbitrary prime and F(x) field of rational functions
+* Roberts' totally twisted Khovanov homology
+* the E^3 page of the twisted spectral sequence Kh(L) => \widehat{HF}(\Sigma_L)
+* spectral sequences of PIDs: lsss over F[x], gss over Z
+* maps induced by cobordisms
+
+5. FOR DEVELOPERS
+
+If you are interested in contributing to knotkit or using it for a new
+application, please contact me: Cotton Seed
+(cotton@math.princeton.edu).  I am interested in developing knotkit to
+be an open platform for performing computations arising in
+low-dimensional topology.
diff --git a/algebra/Q.h b/algebra/Q.h
index b6ee350..21a1d06 100644
--- a/algebra/Q.h
+++ b/algebra/Q.h
@@ -31,16 +31,16 @@ class Q
     Q_impl (reader &r)
     {
       mpq_init (x);
-      mpz_inp_raw (mpq_numref (x), r.fp);
-      mpz_inp_raw (mpq_denref (x), r.fp);
+      r.read_mpz (mpq_numref (x));
+      r.read_mpz (mpq_denref (x));
     }
     
     ~Q_impl () { mpq_clear (x); }
     
     void write_self (writer &w) const
     {
-      mpz_out_raw (w.fp, mpq_numref (x));
-      mpz_out_raw (w.fp, mpq_denref (x));
+      w.write_mpz (mpq_numref (x));
+      w.write_mpz (mpq_denref (x));
     }
   };
   
@@ -163,12 +163,12 @@ class Q
   
   Q div (const Q &d) const { return operator / (d); }
   
-  triple<Q, Q, Q> extended_gcd (const Q &q) const
+  tuple<Q, Q, Q> extended_gcd (const Q &q) const
   {
     if (*this != 0)
-      return triple<Q, Q, Q> (*this, 1, 0);
+      return tuple<Q, Q, Q> (*this, 1, 0);
     else
-      return triple<Q, Q, Q> (q, 0, 1);
+      return tuple<Q, Q, Q> (q, 0, 1);
   }
   
   Q gcd (const Q &q) const
diff --git a/algebra/Z.h b/algebra/Z.h
index 3575992..d23c0fc 100644
--- a/algebra/Z.h
+++ b/algebra/Z.h
@@ -21,14 +21,14 @@ class Z
     Z_impl (reader &r)
     {
       mpz_init (x);
-      mpz_inp_raw (x, r.fp);
+      r.read_mpz (x);
     }
     
     ~Z_impl () { mpz_clear (x); }
     
     void write_self (writer &w) const
     {
-      mpz_out_raw (w.fp, x);
+      w.write_mpz (x);
     }
   };
   
@@ -48,6 +48,7 @@ class Z
   Z &operator = (int x) { impl = new Z_impl (x); return *this; }
   
   bool operator == (const Z &z) const { return mpz_cmp (impl->x, z.impl->x) == 0; }
+  bool operator != (const Z &z) const { return !operator == (z); }
   
   bool operator == (int y) const { return mpz_cmp_si (impl->x, y) == 0; }
   bool operator != (int y) const { return !operator == (y); }
@@ -91,13 +92,13 @@ class Z
     return Z (STEAL, x);
   }
   
-  Z operator / (const Z &z) const
+  Z operator / (const Z &denom) const
   {
-    if (mpz_cmp_si (z.impl->x, 1) == 0)
+    if (mpz_cmp_si (denom.impl->x, 1) == 0)
       return *this;
     else
       {
-	assert (mpz_cmp_si (z.impl->x, -1) == 0);
+	assert (mpz_cmp_si (denom.impl->x, -1) == 0);
 	
 	mpz_t x;
 	mpz_init (x);
@@ -112,6 +113,7 @@ class Z
     return Z (COPY, *this);
   }
   
+  // *this += z1*z2
   Z &muladdeq (const Z &z1, const Z &z2)
   {
     mpz_addmul (impl->x, z1.impl->x, z2.impl->x);
@@ -148,33 +150,30 @@ class Z
     return *this;
   }
   
-  bool divides (const Z &n) const
+  bool divides (const Z &num) const
   {
-    // d = *this
-    return mpz_divisible_p (n.impl->x, impl->x);
+    return mpz_divisible_p (num.impl->x, impl->x);
   }
+  bool operator | (const Z &num) const { return divides (num); }
   
-  bool operator | (const Z &z) const { return divides (z); }
-  
-  Z div (const Z &d) const
+  Z divide_exact (const Z &denom) const
   {
-    // n = *this
+    // num = *this
     mpz_t q;
     mpz_init (q);
-    mpz_divexact (q, impl->x, d.impl->x);
+    mpz_divexact (q, impl->x, denom.impl->x);
     return Z (STEAL, q);
   }
   
-  triple<Z, Z, Z> extended_gcd (const Z &z) const
+  tuple<Z, Z> divide_with_remainder (const Z &denom) const
   {
-    mpz_t d, s, t;
-    mpz_init (d);
-    mpz_init (s);
-    mpz_init (t);
-    mpz_gcdext (d, s, t, impl->x, z.impl->x);
-    return triple<Z, Z, Z> (Z (STEAL, d),
-			    Z (STEAL, s),
-			    Z (STEAL, t));
+    // *this = num
+    mpz_t q, r;
+    mpz_init (q);
+    mpz_init (r);
+    mpz_tdiv_qr (q, r, impl->x, denom.impl->x);
+    return make_tuple (Z (STEAL, q),
+		       Z (STEAL, r));
   }
   
   Z gcd (const Z &z) const
@@ -191,6 +190,18 @@ class Z
     return Z (STEAL, m);
   }
   
+  tuple<Z, Z, Z> extended_gcd (const Z &z) const
+  {
+    mpz_t d, s, t;
+    mpz_init (d);
+    mpz_init (s);
+    mpz_init (t);
+    mpz_gcdext (d, s, t, impl->x, z.impl->x);
+    return make_tuple (Z (STEAL, d),
+		       Z (STEAL, s),
+		       Z (STEAL, t));
+  }
+  
   static void show_ring () { printf ("Z"); }
   void show_self () const { mpz_out_str (stdout, 10, impl->x); }
   void display_self () const { show_self (); newline (); }
diff --git a/algebra/Z2.h b/algebra/Z2.h
index f13f192..09be75a 100644
--- a/algebra/Z2.h
+++ b/algebra/Z2.h
@@ -11,15 +11,18 @@ class Z2
  public:
   Z2 () : v(0) { }
   Z2 (int x) : v((bool)(x & 1)) { }
+  Z2 (unsigned x) : v((bool)(x & 1)) { }
   Z2 (bool v_) : v(v_) { }
   Z2 (const Z2 &x) : v(x.v) { }
   Z2 (copy, const Z2 &x) : v(x.v) { }
+  Z2 (reader &r) { v = r.read_bool (); }
   ~Z2 () { }
   
   Z2 &operator = (const Z2 &x) { v = x.v; return *this; }
   Z2 &operator = (int x) { v = (bool)(x & 1); return *this; }
   
   bool operator == (const Z2 &x) const { return v == x.v; }
+  bool operator != (const Z2 &x) const { return v != x.v; }
   
   bool operator == (int x) const { return v == (bool)(x & 1); }
   bool operator != (int x) const { return !operator == (x); }
@@ -39,6 +42,12 @@ class Z2
   Z2 &operator *= (const Z2 &x) { v &= x.v; return *this; }
   Z2 &operator /= (const Z2 &x) { assert (x.v); return *this; }
   
+  // *this += z1*z2
+  Z2 &muladdeq (const Z2 &z1, const Z2 &z2)
+  {
+    return operator += (z1 * z2);
+  }
+  
   bool divides (Z2 x) const { return v || !x.v; }
   bool operator | (const Z2 x) const { return divides (x); }
   
@@ -54,15 +63,16 @@ class Z2
     return Z2 (1);
   }
   
-  triple<Z2, Z2, Z2> extended_gcd (Z2 x) const
+  tuple<Z2, Z2, Z2> extended_gcd (Z2 x) const
   {
     if (v)
-      return triple<Z2, Z2, Z2> (Z2 (1), Z2 (1), Z2 (0));
+      return make_tuple (Z2 (1), Z2 (1), Z2 (0));
     else
-      return triple<Z2, Z2, Z2> (Z2 (1), Z2 (0), Z2 (1));
+      return make_tuple (Z2 (1), Z2 (0), Z2 (1));
   }
   
   static void show_ring () { printf ("Z2"); }
+  void write_self (writer &w) const { w.write_bool (v); }
   void show_self () const { printf ("%d", (int)v); }
   void display_self () const { printf ("%d\n", (int)v); }
 };
diff --git a/algebra/Zp.h b/algebra/Zp.h
index dc53609..bd3fe0a 100644
--- a/algebra/Zp.h
+++ b/algebra/Zp.h
@@ -32,6 +32,7 @@ class Zp
   }
   
   bool operator == (const Zp &x) const { return v == x.v; }
+  bool operator != (const Zp &x) const { return !operator == (x); }
   
   bool operator == (int x) const { return operator == (Zp (x)); }
   bool operator != (int x) const { return !operator == (x); }
@@ -56,11 +57,11 @@ class Zp
   
   Zp recip () const
   {
-    triple<unsigned, int, int> t = unsigned_extended_gcd (v, p);
-    assert (t.first == 1);
-    assert ((int)t.first == t.second*(int)v  + t.third*(int)p);
+    tuple<unsigned, int, int> t = unsigned_extended_gcd (v, p);
+    assert (get<0> (t) == 1);
+    assert ((int)get<0> (t) == get<1> (t)*(int)v  + get<2> (t)*(int)p);
     
-    return Zp (t.second);
+    return Zp (get<1> (t));
   }
   
   Zp &operator += (const Zp &x)
@@ -92,12 +93,12 @@ class Zp
   
   Zp div (const Zp &d) const { return operator / (d); }
   
-  triple<Zp, Zp, Zp> extended_gcd (const Zp &x) const
+  tuple<Zp, Zp, Zp> extended_gcd (const Zp &x) const
   {
     if (v)
-      return triple<Zp, Zp, Zp> (v, 1, 0);
+      return make_tuple (v, Zp (1), Zp (0));
     else
-      return triple<Zp, Zp, Zp> (x, 0, 1);
+      return make_tuple (x, Zp (0), Zp (1));
   }
   
   Zp gcd (const Zp &x) const
diff --git a/algebra/algebra.cpp b/algebra/algebra.cpp
index e467674..82b01a2 100644
--- a/algebra/algebra.cpp
+++ b/algebra/algebra.cpp
@@ -16,8 +16,8 @@ unsigned_gcd (unsigned a, unsigned b)
 unsigned
 int_gcd (int a, int b)
 {
-  return unsigned_gcd ((unsigned)std::abs (a),
-		       (unsigned)std::abs (b));
+  return unsigned_gcd ((unsigned)abs (a),
+		       (unsigned)abs (b));
 }
 
 uint64
@@ -35,47 +35,47 @@ uint64_gcd (uint64 a, uint64 b)
 uint64
 int64_gcd (int64 a, int64 b)
 {
-  return uint64_gcd ((uint64)std::abs (a),
-		     (uint64)std::abs (b));
+  return uint64_gcd ((uint64)abs64 (a),
+		     (uint64)abs64 (b));
 }
 
-static triple<unsigned, int, int>
+static tuple<unsigned, int, int>
 extended_gcd_1 (int a, int b)
 {
   if (b == 0)
-    return triple<unsigned, int, int> (a, 1, 0);
+    return tuple<unsigned, int, int> (a, 1, 0);
   
   unsigned t = a % b;
   if (!t)
-    return triple<unsigned, int, int> (b, 0, 1);
+    return make_tuple (b, 0, 1);
   else
     {
-      triple<unsigned, int, int> s = unsigned_extended_gcd (b, t);
+      tuple<unsigned, int, int> s = unsigned_extended_gcd (b, t);
       
-      unsigned d = s.first;
-      int x = s.third,
-	y = s.second - s.third * (a / b);
+      unsigned d = get<0> (s);
+      int x = get<2> (s),
+	y = get<1> (s) - get<2> (s) * (a / b);
       
       assert ((int)d == a*x + b*y);
       
-      return triple<unsigned, int, int> (d, x, y);
+      return make_tuple (d, x, y);
     }
 }
 
-triple<unsigned, int, int>
+tuple<unsigned, int, int>
 unsigned_extended_gcd (unsigned a, unsigned b)
 {
   return extended_gcd_1 ((int)a, (int)b);
 }
 
-triple<unsigned, int, int>
+tuple<unsigned, int, int>
 extended_gcd (int a, int b)
 {
-  triple<unsigned, int, int> t = extended_gcd_1 (std::abs (a),
-						 std::abs (b));
-  unsigned d = t.first;
-  int x = t.second,
-    y = t.third;
+  tuple<unsigned, int, int> t = extended_gcd_1 (abs (a),
+						abs (b));
+  unsigned d = get<0> (t);
+  int x = get<1> (t),
+    y = get<2> (t);
   if (a < 0)
     x *= -1;
   if (b < 0)
@@ -83,7 +83,7 @@ extended_gcd (int a, int b)
   
   assert ((int)d == a*x + b*y);
   
-  return triple<unsigned, int, int> (d, x, y);
+  return make_tuple (d, x, y);
 }
 
 unsigned
@@ -100,12 +100,12 @@ uint64_lcm (uint64 a, uint64 b)
 
 unsigned int_lcm (int a, int b)
 {
-  return unsigned_lcm ((unsigned)std::abs (a),
-		       (unsigned)std::abs (b));
+  return unsigned_lcm ((unsigned)abs (a),
+		       (unsigned)abs (b));
 }
 
 uint64 int64_lcm (int64 a, int64 b)
 {
-  return uint64_lcm ((uint64)std::abs (a),
-		     (uint64)std::abs (b));
+  return uint64_lcm ((uint64)abs64 (a),
+		     (uint64)abs64 (b));
 }
diff --git a/algebra/algebra.h b/algebra/algebra.h
index 42319f5..4acd0ca 100644
--- a/algebra/algebra.h
+++ b/algebra/algebra.h
@@ -1,6 +1,4 @@
 
-#include <gmp.h>
-
 #include <lib/lib.h>
 
 inline int recip (int x)
@@ -42,9 +40,9 @@ unsigned int_lcm (int a, int b);
 uint64 int64_lcm (int64 a, int64 b);
 
 // (d, x, y) = gcd (a, b) where x*a + y*b = d
-triple<unsigned, int, int> extended_gcd (int a, int b);
+tuple<unsigned, int, int> extended_gcd (int a, int b);
 
-triple<unsigned, int, int> unsigned_extended_gcd (unsigned a, unsigned b);
+tuple<unsigned, int, int> unsigned_extended_gcd (unsigned a, unsigned b);
 
 template<class R> class linear_combination;
 template<class R> class linear_combination_const_iter;
diff --git a/algebra/fraction_field.h b/algebra/fraction_field.h
index d2ea6a4..3f648b6 100644
--- a/algebra/fraction_field.h
+++ b/algebra/fraction_field.h
@@ -36,6 +36,7 @@ template<class T> class fraction_field
   fraction_field &operator = (int x) { num = x; denom = 1; return *this; }
   
   bool operator == (const fraction_field &q) const { return num * q.denom == q.num * denom; }
+  bool operator != (const fraction_field &q) const { return !operator == (q); }
   
   bool operator == (int x) const { return num == denom * T (x); }
   bool operator != (int x) const { return !operator == (x); }
@@ -66,6 +67,11 @@ template<class T> class fraction_field
     return fraction_field (new_num, denom*xe);
   }
   
+  fraction_field operator - () const
+  {
+    return fraction_field (-num, denom);
+  }
+  
   fraction_field operator * (const fraction_field &x) const
   {
     T d1 = num.gcd (x.denom);
@@ -187,12 +193,12 @@ template<class T> class fraction_field
     return fraction_field (1);
   }
   
-  triple<fraction_field, fraction_field, fraction_field> extended_gcd (const fraction_field &x) const
+  tuple<fraction_field, fraction_field, fraction_field> extended_gcd (const fraction_field &x) const
   {
     if (*this != 0)
-      return triple<fraction_field, fraction_field, fraction_field> (*this, 1, 0);
+      return make_tuple (*this, fraction_field (1), fraction_field (0));
     else
-      return triple<fraction_field, fraction_field, fraction_field> (x, 0, 1);
+      return make_tuple (x, fraction_field (0), fraction_field (1));
   }
   
   static void show_ring () { printf ("fraction_field("); T::show_ring (); printf (")"); }
@@ -210,7 +216,6 @@ fraction_field<T>::reduce ()
     }
   
   T d = num.gcd (denom);
-  
   num = num.divide_exact (d);
   denom = denom.divide_exact (d);
 }
@@ -220,7 +225,6 @@ fraction_field<T>::check ()
 {
   if (num == 0)
     return;
-  // check denom == 1
   
   // assert (num.gcd (denom) == 1);
 }
diff --git a/algebra/grading.h b/algebra/grading.h
index 936aeb3..cbd3972 100644
--- a/algebra/grading.h
+++ b/algebra/grading.h
@@ -8,6 +8,12 @@ class grading
   grading () : h(0), q(0) { }
   grading (int h_, int q_) : h(h_), q(q_) { }
   grading (const grading &gr) : h(gr.h), q(gr.q) { }
+  grading (reader &r)
+  {
+    h = r.read_int ();
+    q = r.read_int ();
+  }
+  
   ~grading () { }
   
   grading &operator = (const grading &gr) { h = gr.h; q = gr.q; return *this; }
@@ -36,6 +42,8 @@ class grading
   }
   
   bool operator == (const grading &gr) const { return h == gr.h && q == gr.q; }
+  bool operator != (const grading &gr) const { return !operator == (gr); }
+  
   bool operator < (const grading &gr) const
   {
     return h < gr.h
@@ -43,7 +51,13 @@ class grading
   }
   
   grading mirror_grading (unsigned nplus, unsigned nminus, bool torsion) const;
-  
+
+  void write_self (writer &w) const
+  {
+    w.write_int (h);
+    w.write_int (q);
+  }
+    
   void show_self () const;
   void display_self () const;
 };
diff --git a/algebra/linear_combination.h b/algebra/linear_combination.h
index dda27ce..373a0b8 100644
--- a/algebra/linear_combination.h
+++ b/algebra/linear_combination.h
@@ -39,8 +39,15 @@ class linear_combination
     for (typename map<unsigned, R>::const_iter i = lc.v; i; i ++)
       v.push (i.key (), R (COPY, i.val ()));
   }
+  
+  linear_combination (reader &r)
+  {
+    m = r.read_mod<R> ();
+    v = map<unsigned, R> (r);
+  }
+  
   ~linear_combination () { }
-
+  
   linear_combination &operator = (const linear_combination &lc)
   {
     m = lc.m;
@@ -55,6 +62,7 @@ class linear_combination
     lc3 -= lc2;
     return lc3 == 0;
   }
+  bool operator != (const linear_combination &lc) const { return !operator == (lc); }
   
   bool operator == (int x) const
   {
@@ -163,6 +171,12 @@ class linear_combination
       assert (!m->is_zero (i.val (), i.key ()));
   }
 #endif
+
+  void write_self (writer &w) const
+  {
+    write (w, *m);
+    write (w, v);
+  }
   
   void show_self () const;
   void display_self () const { show_self (); newline (); }
@@ -336,6 +350,12 @@ class linear_combination<Z2>
   linear_combination (ptr<const Z2mod> m_) : m(m_) { }
   linear_combination (const linear_combination &lc) : m(lc.m), v(lc.v) { }
   linear_combination (copy, const linear_combination &lc) : m(lc.m), v(COPY, lc.v) { }
+  linear_combination (reader &r)
+  {
+    m = r.read_mod<Z2> ();
+    v = set<unsigned> (r);
+  }
+  
   ~linear_combination () { }
   
   linear_combination &operator = (const linear_combination &lc)
@@ -346,6 +366,7 @@ class linear_combination<Z2>
   }
   
   bool operator == (const linear_combination &lc) const { assert (m == lc.m); return v == lc.v; }
+  bool operator != (const linear_combination &lc) const { return !operator == (lc); }
   
   bool operator == (int x) const { assert (x == 0); return v.is_empty (); }
   bool operator != (int x) const { return !operator == (x); }
@@ -445,6 +466,12 @@ class linear_combination<Z2>
   void check () const;
 #endif
   
+  void write_self (writer &w) const
+  {
+    write (w, *m);
+    write (w, v);
+  }
+  
   void show_self () const;
   void display_self () const { show_self (); newline (); }
 };
diff --git a/algebra/module.h b/algebra/module.h
index 610802b..acd217b 100644
--- a/algebra/module.h
+++ b/algebra/module.h
@@ -14,10 +14,15 @@ template<class R>
 class module : public refcounted
 {
  private:
+  friend class reader;
+  friend class writer;
+  
   unsigned id;
   
   static unsigned id_counter;
   
+  static map<unsigned, ptr<const module<R> > > reader_id_module;
+  
   static map<basedvector<unsigned, 1>,
     ptr<const direct_sum<R> > > direct_sum_idx;
   static map<basedvector<unsigned, 1>,
@@ -28,13 +33,13 @@ class module : public refcounted
  public:
   module ()
   {
-    id = id_counter;
     id_counter ++;
+    id = id_counter;
   }
-  module (const module &); // doesn't exist
+  module (const module &) = delete;
   virtual ~module () { }
   
-  module &operator = (const module &); // doesn't exist
+  module &operator = (const module &) = delete;
   
  public:
   // the number of generators; n
@@ -48,6 +53,9 @@ class module : public refcounted
   
   // r < i <= n
   virtual R generator_ann (unsigned i) const = 0;
+
+  basedvector<grading, 1> grading_vector () const;
+  set<grading> gradings () const;
   
   bool is_free () const { return dim () == free_rank (); }
   
@@ -56,7 +64,12 @@ class module : public refcounted
     if (i <= free_rank ())
       return c == 0;
     else
-      return generator_ann (i) | c;
+      {
+	// ???
+	abort ();
+	
+	// return generator_ann (i) | c;
+      }
   }
   
   R annihilator (R c, unsigned i) const
@@ -75,6 +88,7 @@ class module : public refcounted
   
   multivariate_laurentpoly<Z> free_poincare_polynomial () const;
   multivariate_laurentpoly<Z> free_delta_poincare_polynomial () const;
+  multivariate_laurentpoly<Z> free_ell_poincare_polynomial () const;
   
   ptr<const direct_sum<R> >
     add (basedvector<ptr<const module<R> >, 1> compound_summands) const;
@@ -143,11 +157,16 @@ class module : public refcounted
     factors.append (this);
   }
   
+  ptr<const free_submodule<R> > graded_piece (grading hq) const;
+  
+  void write_self (writer &w) const { w.write_mod<R> (this); }
   void show_self () const;
   void display_self () const;
 };
 
-template<class R> unsigned module<R>::id_counter = 1;
+template<class R> unsigned module<R>::id_counter = 0;
+
+template<class R> map<unsigned, ptr<const module<R> > > module<R>::reader_id_module;
 
 template<class R> map<basedvector<unsigned, 1>,
   ptr<const direct_sum<R> > > module<R>::direct_sum_idx;
@@ -466,11 +485,11 @@ class base_module : public module<R>
   G g;
   
  public:
-  base_module (); // doesn't exist
+  base_module () = delete;
   base_module (const G &g_) : g(g_) { }
   ~base_module () { }
   
-  base_module &operator = (const base_module &); // doesn't exist
+  base_module &operator = (const base_module &) = delete;
   
   unsigned dim () const { return g.dim (); }
   unsigned free_rank () const { return g.free_rank (); }
@@ -487,17 +506,19 @@ class explicit_module : public module<R>
   basedvector<grading, 1> hq;
   
  public:
-  explicit_module (); // doesn't exist
+  explicit_module () = delete;
   explicit_module (unsigned r_,
 		   basedvector<R, 1> ann_,
 		   basedvector<grading, 1> hq_)
     : r(r_), ann(ann_), hq(hq_)
-  { }
+  {
+    assert (hq.size () == r + ann.size ());
+  }
   
   explicit explicit_module (unsigned r_, basedvector<grading, 1> hq_) : r(r_), hq(hq_) { }
   ~explicit_module () { }
   
-  explicit_module &operator = (const explicit_module &); // doesn't exist
+  explicit_module &operator = (const explicit_module &) = delete;
   
   unsigned dim () const { return r + ann.size (); }
   unsigned free_rank () const { return r; }
@@ -525,7 +546,7 @@ class free_submodule : public module<R>
   { }
   ~free_submodule () { }
   
-  free_submodule &operator = (const free_submodule &); // doesn't exist
+  free_submodule &operator = (const free_submodule &) = delete;
   
   ptr<const module<R> > parent_module () const { return parent; }
   
@@ -542,6 +563,9 @@ class free_submodule : public module<R>
   
   linear_combination<R> restrict (linear_combination<R> v0) const;
   ptr<const free_submodule<R> > restrict_submodule (ptr<const free_submodule<R> > m) const;
+  
+  ptr<const free_submodule<R> > intersection (ptr<const free_submodule<R> > m) const;
+  ptr<const free_submodule<R> > plus (ptr<const free_submodule<R> > m) const;
 };
 
 template<class R> 
@@ -562,13 +586,13 @@ class quotient_module : public module<R>
   basedvector<map<unsigned, R>, 1> pi;
   
  public:
-  quotient_module (const quotient_module &); // doesn't exist
+  quotient_module (const quotient_module &) = delete;
   quotient_module (ptr<const module<R> > parent_)
     : parent(parent_)
   { }
   ~quotient_module () { }
   
-  quotient_module &operator = (const quotient_module &); // doesn't exsit
+  quotient_module &operator = (const quotient_module &) = delete;
   
   ptr<const module<R> > parent_module () const { return parent; }
   
@@ -617,7 +641,7 @@ class map_impl : public refcounted
   ptr<const module<R> > to;
   
  public:
-  map_impl (const map_impl &); // doesn't exist
+  map_impl (const map_impl &) = delete;
   map_impl (ptr<const module<R> > fromto)
     : from(fromto), to(fromto)
   { }
@@ -626,9 +650,10 @@ class map_impl : public refcounted
   { }
   virtual ~map_impl () { }
   
-  map_impl &operator = (const map_impl &); // doesn't exist
+  map_impl &operator = (const map_impl &) = delete;
   
-  virtual linear_combination<R> column (unsigned i) const = 0;
+  virtual const linear_combination<R> column (unsigned i) const = 0;
+  virtual const linear_combination<R> column_copy (unsigned i) const { return column (i); }
   
   linear_combination<R> map (const linear_combination<R> &lc) const
   {
@@ -657,7 +682,11 @@ class explicit_map_impl : public map_impl<R>
   { }
   ~explicit_map_impl () { }
   
-  linear_combination<R> column (unsigned i) const { return columns[i]; }
+  const linear_combination<R> column (unsigned i) const { return columns[i]; }
+  const linear_combination<R> column_copy (unsigned i) const
+  {
+    return linear_combination<R> (COPY, columns[i]);
+  }
 };
 
 template<class R>
@@ -667,7 +696,7 @@ class zero_map_impl : public map_impl<R>
   zero_map_impl (ptr<const module<R> > fromto) : map_impl<R>(fromto) { }
   zero_map_impl (ptr<const module<R> > from, ptr<const module<R> > to) : map_impl<R>(from, to) { }
   
-  linear_combination<R> column (unsigned i) const { return linear_combination<R> (this->to); }
+  const linear_combination<R> column (unsigned i) const { return linear_combination<R> (this->to); }
 };
 
 template<class R>
@@ -677,7 +706,7 @@ class id_map_impl : public map_impl<R>
   id_map_impl (ptr<const module<R> > fromto) : map_impl<R>(fromto) { }
   id_map_impl (ptr<const module<R> > from, ptr<const module<R> > to) : map_impl<R>(from, to) { }
   
-  linear_combination<R> column (unsigned i) const
+  const linear_combination<R> column (unsigned i) const
   {
     linear_combination<R> r (this->to);
     r.muladd (1, i);
@@ -700,7 +729,7 @@ class composition_impl : public map_impl<R>
     assert (g->to == f->from);
   }
   
-  linear_combination<R> column (unsigned i) const
+  const linear_combination<R> column (unsigned i) const
   {
     return f->map (g->column (i));
   }
@@ -721,7 +750,7 @@ class direct_sum_impl : public map_impl<R>
   {
   }
   
-  linear_combination<R> column (unsigned i) const
+  const linear_combination<R> column (unsigned i) const
   {
     pair<unsigned, unsigned> p = f->from->project (g->from, i);
     
@@ -756,7 +785,7 @@ class tensor_impl : public map_impl<R>
   {
   }
   
-  linear_combination<R> column (unsigned i) const
+  const linear_combination<R> column (unsigned i) const
   {
     pair<unsigned, unsigned> p = f->from->generator_factors (g->from, i);
     
@@ -849,6 +878,15 @@ class mod_map
   mod_map (const map_builder<R> &b)
     : impl(new explicit_map_impl<R> (b.from, b.to, b.columns))
   { }
+  
+  mod_map (reader &r)
+  {
+    ptr<const module<R> > from = r.read_mod<R> ();
+    ptr<const module<R> > to = r.read_mod<R> ();
+    basedvector<linear_combination<R>, 1> columns (r);
+    impl = new explicit_map_impl<R> (from, to, columns);
+  }
+  
   ~mod_map () { }
   
   mod_map &operator = (const mod_map &m) { impl = m.impl; return *this; }
@@ -867,6 +905,7 @@ class mod_map
       }
     return 1;
   }
+  bool operator != (const mod_map &m) const { return !operator == (m); }
   
   bool operator == (int x) const
   {
@@ -883,8 +922,10 @@ class mod_map
   
   bool operator != (int x) const { return !operator == (x); }
   
-  linear_combination<R> column (unsigned i) const { return impl->column (i); }
-  linear_combination<R> operator [] (unsigned i) const { return impl->column (i); }
+  const linear_combination<R> column (unsigned i) const { return impl->column (i); }
+  const linear_combination<R> operator [] (unsigned i) const { return impl->column (i); }
+  
+  const linear_combination<R> column_copy (unsigned i) const { return impl->column_copy (i); }
   
   linear_combination<R> map (const linear_combination<R> &lc) const { return impl->map (lc); }
   mod_map compose (const mod_map &m) const
@@ -908,6 +949,7 @@ class mod_map
   
   // ?? add and other map operations should not be explicit
   mod_map operator + (const mod_map &m) const;
+  mod_map operator * (const R &c) const;
   
   bool homogeneous () const;
   void check_grading (grading delta) const;
@@ -917,6 +959,7 @@ class mod_map
   
   // inj : im -> to
   ptr<const free_submodule<R> > image () const;
+  ptr<const free_submodule<R> > image (basedvector<linear_combination<R>, 1> vs) const;
   
   ptr<const quotient_module<R> > cokernel () const;
   
@@ -934,9 +977,19 @@ class mod_map
   mod_map induced_map_to (ptr<const quotient_module<R> > new_to);
   mod_map induced_map (ptr<const quotient_module<R> > new_fromto);
   
+  mod_map graded_piece (grading hq) const;
+  
   // ???
   basedvector<linear_combination<R>, 1> explicit_columns () const;
   
+  void write_self (writer &w) const
+  {
+    // write explicitly
+    write (w, *impl->from);
+    write (w, *impl->to);
+    write (w, explicit_columns ());
+  }
+  
   void show_self () const;
   void display_self () const;
 };
@@ -1095,16 +1148,16 @@ quotient_helper<R>::improve_pivot_row (unsigned i, unsigned j, unsigned i2)
     }
 #endif
   
-  triple<R, R, R> t = rc.extended_gcd (r2c);
-  assert (t.first == rc*t.second + t.third*r2c);
+  tuple<R, R, R> t = rc.extended_gcd (r2c);
+  assert (get<0> (t) == rc*get<1> (t) + get<2> (t)*r2c);
   
-  rows[i] = r*t.second + r2*t.third;
-  rows[i2] = (rc.div (t.first))*r2 - (r2c.div (t.first))*r;
+  rows[i] = r*get<1> (t) + r2*get<2> (t);
+  rows[i2] = (rc.div (get<0> (t)))*r2 - (r2c.div (get<0> (t)))*r;
   
-  assert ((rc | r2c) == rc.divides (t.first));
-  assert (!rc.divides (t.first) || rows[i2](j) == 0);
+  assert ((rc | r2c) == rc.divides (get<0> (t)));
+  assert (!rc.divides (get<0> (t)) || rows[i2](j) == 0);
   
-  return !rc.divides (t.first);
+  return !rc.divides (get<0> (t));
 }
 
 template<class R> bool
@@ -1149,8 +1202,8 @@ quotient_helper<R>::improve_pivot_column (unsigned i, unsigned j, unsigned j2)
     }
 #endif
   
-  triple<R, R, R> t = rc.extended_gcd (rc2);
-  assert (t.first == rc*t.second + t.third*rc2);
+  tuple<R, R, R> t = rc.extended_gcd (rc2);
+  assert (get<0> (t) == rc*get<1> (t) + get<2> (t)*rc2);
   
   for (unsigned k = 1; k <= rows.size (); k ++)
     {
@@ -1158,9 +1211,9 @@ quotient_helper<R>::improve_pivot_column (unsigned i, unsigned j, unsigned j2)
       R rkc = rk(j),
 	rkc2 = rk(j2);
       
-      rk.set_coeff (rkc*t.second + rkc2*t.third,
+      rk.set_coeff (rkc*get<1> (t) + rkc2*get<2> (t),
 		    j);
-      rk.set_coeff (rkc2*(rc.div (t.first)) - rkc*(rc2.div (t.first)),
+      rk.set_coeff (rkc2*(rc.div (get<0> (t))) - rkc*(rc2.div (get<0> (t))),
 		    j2);
     }
   
@@ -1169,8 +1222,8 @@ quotient_helper<R>::improve_pivot_column (unsigned i, unsigned j, unsigned j2)
   
   assert (g.hq () == g2.hq ());
   
-  generators[j] = (rc.div (t.first)) * g + (rc2.div (t.first)) * g2;
-  generators[j2] = t.second * g2 - t.third * g;
+  generators[j] = (rc.div (get<0> (t))) * g + (rc2.div (get<0> (t))) * g2;
+  generators[j2] = get<1> (t) * g2 - get<2> (t) * g;
   
  #if 0
   for (unsigned k = 1; k <= rows.size (); k ++)
@@ -1189,8 +1242,8 @@ quotient_helper<R>::improve_pivot_column (unsigned i, unsigned j, unsigned j2)
       R d = ginv(j),
 	d2 = ginv(j2);
       
-      ginv.set_coeff (t.second*d + t.third*d2, j);
-      ginv.set_coeff (rc.div (t.first) * d2 - rc2.div (t.first) * d, j2);
+      ginv.set_coeff (get<1> (t)*d + get<2> (t)*d2, j);
+      ginv.set_coeff (rc.div (get<0> (t)) * d2 - rc2.div (get<0> (t)) * d, j2);
     }
   
 #if 0
@@ -1207,10 +1260,10 @@ quotient_helper<R>::improve_pivot_column (unsigned i, unsigned j, unsigned j2)
     }
 #endif
   
-  assert ((rc | rc2) == rc.divides (t.first));
-  assert (!rc.divides (t.first) || r(j2) == 0);
+  assert ((rc | rc2) == rc.divides (get<0> (t)));
+  assert (!rc.divides (get<0> (t)) || r(j2) == 0);
   
-  return !rc.divides (t.first);
+  return !rc.divides (get<0> (t));
 }
 
 template<class R> void
@@ -1437,6 +1490,58 @@ module<R>::free_delta_poincare_polynomial () const
   return r;
 }
 
+template<class R> multivariate_laurentpoly<Z>
+module<R>::free_ell_poincare_polynomial () const
+{
+  multivariate_laurentpoly<Z> r;
+  for (unsigned i = 1; i <= free_rank (); i ++)
+    {
+      grading hq = generator_grading (i);
+      multivariate_laurent_monomial m;
+      m.push_exponent (1, hq.h - hq.q);
+      r.muladdeq (1, m);
+    }
+  return r;
+}
+
+template<class R> basedvector<grading, 1>
+module<R>::grading_vector () const
+{
+  basedvector<grading, 1> v (dim ());
+  for (unsigned i = 1; i <= dim (); i ++)
+    v[i] = generator_grading (i);
+  return v;
+}
+
+template<class R> set<grading> 
+module<R>::gradings () const
+{
+  set<grading> gs;
+  for (unsigned i = 1; i <= dim (); i ++)
+    gs += generator_grading (i);
+  return gs;
+}
+
+template<class R> ptr<const free_submodule<R> > 
+module<R>::graded_piece (grading hq) const
+{
+  basedvector<linear_combination<R>, 1> s;
+  for (unsigned i = 1; i <= dim (); i ++)
+    {
+      grading ihq = generator_grading (i);
+      if (ihq.h == hq.h
+	  && ihq.q == hq.q)
+	{
+	  linear_combination<R> v (this);
+	  v.muladd (1, i);
+	  s.append (v);
+	}
+    }
+  
+  mod_span<R> span (this, s);
+  return submodule (span);
+}
+
 template<class R> void
 module<R>::show_self () const
 {
@@ -1520,6 +1625,28 @@ mod_map<R>::induced_map (ptr<const quotient_module<R> > new_fromto)
   return new explicit_map_impl<R> (new_fromto, v);
 }
 
+template<class R> mod_map<R>
+mod_map<R>::graded_piece (grading hq) const
+{
+  basedvector<linear_combination<R>, 1> v (impl->from->dim ());
+  for (unsigned i = 1; i <= impl->from->dim (); i ++)
+    {
+      grading ihq = impl->from->generator_grading (i);
+      
+      linear_combination<R> c = column (i);
+      linear_combination<R> d (impl->to);
+      for (linear_combination_const_iter<R> j = c; j; j ++)
+	{
+	  grading jhq = impl->from->generator_grading (j.key ());
+	  if (jhq.h - ihq.h == hq.h
+	      && jhq.q - ihq.q == hq.q)
+	    d.muladd (j.val (), j.key ());
+	}
+      v[i] = d;
+    }
+  return mod_map (IMPL, new explicit_map_impl<R> (impl->from, impl->to, v));
+}
+
 template<class R> mod_map<R> 
 mod_map<R>::restrict_from (ptr<const free_submodule<R> > new_from) const
 {
@@ -1528,7 +1655,7 @@ mod_map<R>::restrict_from (ptr<const free_submodule<R> > new_from) const
   basedvector<linear_combination<R>, 1> v (new_from->dim ());
   for (unsigned i = 1; i <= new_from->dim (); i ++)
     v[i] = map (new_from->inject_generator (i));
-  return new explicit_map_impl<R> (new_from, impl->to, v);
+  return mod_map (IMPL, new explicit_map_impl<R> (new_from, impl->to, v));
 }
 
 template<class R> mod_map<R> 
@@ -1552,7 +1679,7 @@ mod_map<R>::restrict (ptr<const free_submodule<R> > new_from,
   basedvector<linear_combination<R>, 1> v (new_from->dim ());
   for (unsigned i = 1; i <= new_from->dim (); i ++)
     v[i] = new_to->restrict (map (new_from->inject_generator (i)));
-  return new explicit_map_impl<R> (new_from, new_to, v);
+  return mod_map (IMPL, new explicit_map_impl<R> (new_from, new_to, v));
 }
 
 template<class R> linear_combination<R>
@@ -1595,7 +1722,94 @@ free_submodule<R>::restrict_submodule (ptr<const free_submodule<R> > m) const
     span[i] = restrict (m->inject_generator (i));
   
   mod_span<R> span2 (this, span);
-  return submodule (span2);
+  return this->submodule (span2);
+}
+
+template<class R> ptr<const free_submodule<R> >
+free_submodule<R>::intersection (ptr<const free_submodule<R> > m) const
+{
+  assert (parent == m->parent);
+  
+  unsigned md = m->dim (),
+    d = dim ();
+  
+  basedvector<linear_combination<R>, 1> intr;
+  
+  basedvector<linear_combination<R>, 1> hperp,
+    hproj;
+  basedvector<unsigned, 1> hpivots;
+  for (unsigned i = 1; i <= md; i ++)
+    {
+      linear_combination<R> perp (COPY, m->gens[i]),
+	proj (parent);
+      
+      for (unsigned j = 1; j <= d; j ++)
+	{
+	  unsigned k = pivots[j];
+	  if (perp % k)
+	    {
+	      const linear_combination<R> &g = gens[j];
+	      R c = g(k);
+	      R d = perp(k);
+	      
+	      assert (c | d);
+	      R q = d.div (c);
+	      
+	      perp.mulsub (q, g);
+	      proj.mulsub (q, g);
+	      
+	      assert (! (perp % k));
+	    }
+	}
+      
+      for (unsigned j = 1; j <= hpivots.size (); j ++)
+	{
+	  unsigned k = hpivots[j];
+	  if (perp % k)
+	    {
+	      const linear_combination<R> &h = hperp[j];
+	      R c = h(k);
+	      R d = perp(k);
+	      
+	      assert (c | d);
+	      R q = d.div (c);
+	      
+	      perp.mulsub (q, h);
+	      proj.mulsub (q, hproj[j]);
+	      
+	      assert (! (perp % k));
+	    }
+	  
+	}
+      
+      if (perp == 0)
+	intr.append (proj);
+      else
+	{
+	  hperp.append (perp);
+	  hproj.append (proj);
+	  hpivots.append (perp.head ().first);
+	}
+    }
+  
+  mod_span<R> span (parent, intr);
+  return parent->submodule (span);
+}
+
+template<class R> ptr<const free_submodule<R> >
+free_submodule<R>::plus (ptr<const free_submodule<R> > m) const
+{
+  assert (parent == m->parent);
+  
+  basedvector<linear_combination<R>, 1> s;
+  for (unsigned i = 1; i <= dim (); i ++)
+    s.append (gens[i]);
+  
+  for (unsigned i = 1; i <= m->dim (); i ++)
+    s.append (m->gens[i]);
+  
+  mod_span<R> span (parent, s);
+  return parent->submodule (span);
 }
 
 template<class R> bool
@@ -1628,6 +1842,14 @@ mod_map<R>::check_grading (grading delta) const
     }
 }
 
+template<class R> mod_map<R>
+mod_map<R>::operator * (const R &c) const
+{
+  basedvector<linear_combination<R>, 1> v (impl->from->dim ());
+  for (unsigned i = 1; i <= impl->from->dim (); i ++)
+    v[i] = c*column (i);
+  return mod_map (IMPL, new explicit_map_impl<R> (impl->from, impl->to, v));
+}
 
 template<class R> mod_map<R>
 mod_map<R>::operator + (const mod_map &m) const
@@ -1636,8 +1858,8 @@ mod_map<R>::operator + (const mod_map &m) const
   
   basedvector<linear_combination<R>, 1> v (impl->from->dim ());
   for (unsigned i = 1; i <= m.impl->from->dim (); i ++)
-    v[i] = column (i) + m.columns (i);
-  return explicit_map_impl<R> (impl->from, impl->to, v);
+    v[i] = column (i) + m.column (i);
+  return mod_map (IMPL, new explicit_map_impl<R> (impl->from, impl->to, v));
 }
 
 template<class R> ptr<const free_submodule<R> > 
@@ -1647,7 +1869,7 @@ mod_map<R>::kernel () const
     to = impl->to;
   
   basedvector<linear_combination<R>, 1> from_xs (from->dim ());
-  for (unsigned i = 1; i <= to->dim (); i ++)
+  for (unsigned i = 1; i <= from->dim (); i ++)
     {
       linear_combination<R> x (from);
       x.muladd (1, i);
@@ -1670,13 +1892,21 @@ mod_map<R>::kernel () const
 	  linear_combination<R> &to_x = to_xs[j],
 	    &from_x = from_xs[j];
 	  R to_xc = to_x(i);
-	  if (! (to_vc | to_xc))
+	  if (to_xc == 0)
+	    continue;
+	  
+	  if (to_vc == 0 && to_xc != 0)
 	    {
-	      triple<R, R, R> t = to_vc.extended_gcd (to_xc);
-	      assert (t.first == to_vc*t.second + t.third*to_xc);
+	      to_v += to_x;
+	      from_v += from_x;
+	    }
+	  else if (! (to_vc | to_xc))
+	    {
+	      tuple<R, R, R> t = to_vc.extended_gcd (to_xc);
+	      assert (get<0> (t) == to_vc*get<1> (t) + get<2> (t)*to_xc);
 	      
-	      to_v = t.second*to_v + t.third*to_x;
-	      from_v = t.second*from_v + t.third*from_x;
+	      to_v = get<1> (t)*to_v + get<2> (t)*to_x;
+	      from_v = get<1> (t)*from_v + get<2> (t)*from_x;
 	      
 	      assert (to_v(i) != 0);
 	    }
@@ -1715,6 +1945,15 @@ mod_map<R>::image () const
   return impl->to->submodule (span);
 }
 
+template<class R> ptr<const free_submodule<R> >
+mod_map<R>::image (basedvector<linear_combination<R>, 1> vs) const
+{
+  mod_span<R> span (impl->from, vs);
+  ptr<const free_submodule<R> > s = impl->from->submodule (span);
+  mod_map<R> r = restrict_from (s);
+  return r.image ();
+}
+
 template<class R> ptr<const quotient_module<R> > 
 mod_map<R>::cokernel () const
 {
@@ -1767,13 +2006,19 @@ mod_span<R>::mod_span (ptr<const module<R> > mod,
 	  
 	  linear_combination<R> &x = xs[j];
 	  R xc = x(i);
+	  if (xc == 0)
+	    continue;
 	  
-	  if (! (vc | xc))
+	  if (vc == 0)
 	    {
-	      triple<R, R, R> t = vc.extended_gcd (xc);
-	      assert (t.first == vc*t.second + t.third*xc);
+	      v += x;
+	    }
+	  else if (! (vc | xc))
+	    {
+	      tuple<R, R, R> t = vc.extended_gcd (xc);
+	      assert (get<0> (t) == vc*get<1> (t) + get<2> (t)*xc);
 	      
-	      v = t.second*v + t.third*x;
+	      v = get<1> (t)*v + get<2> (t)*x;
 	      
 	      assert (v(i) != 0);
 	    }
@@ -1820,8 +2065,68 @@ mod_map<R>::display_self () const
     {
       printf ("  %d ", i);
       impl->from->show_generator (i);
+      printf (" ");
+      show (impl->from->generator_grading (i));
       printf (": ");
       show (column (i));
       newline ();
     }
 }
+
+// ??? io
+
+template<class R> void
+writer::write_mod (ptr<const module<R> > m)
+{
+  pair<unsigned &, bool> p = aw->id_io_id.find (m->id);
+  if (p.second)
+    {
+      write_int ((int)p.first);
+    }
+  else
+    {
+      ++ aw->io_id_counter;
+      unsigned io_id = aw->io_id_counter;
+      
+      p.first = io_id;
+      
+      write_int (- (int)io_id);
+      
+      unsigned n = m->dim (),
+	r = m->free_rank ();
+      write_unsigned (n);
+      write_unsigned (r);
+      for (unsigned i = 1; i <= n; i ++)
+	write (*this, m->generator_grading (i));
+      for (unsigned i = r + 1; i <= n; i ++)
+	write (*this, m->generator_ann (i));
+    }
+}
+
+template<class R> ptr<const module<R> > 
+reader::read_mod ()
+{
+  int io_id = read_int ();
+  if (io_id < 0)
+    {
+      unsigned n = read_unsigned ();
+      unsigned r = read_unsigned ();
+      basedvector<grading, 1> gr (n);
+      for (unsigned i = 1; i <= n; i ++)
+	gr[i] = grading (*this);
+      basedvector<R, 1> ann (n - r);
+      for (unsigned i = r + 1; i <= n; i ++)
+	ann[i - r] = R (*this);
+      
+      ptr<const module<R> > m = new explicit_module<R> (r, ann, gr);
+      ar->io_id_id.push ((unsigned)(-io_id), m->id);
+      module<R>::reader_id_module.push (m->id, m);
+      
+      return m;
+    }
+  else
+    {
+      unsigned id = ar->io_id_id(io_id);
+      return module<R>::reader_id_module(id);
+    }
+}
diff --git a/algebra/multivariate_laurentpoly.h b/algebra/multivariate_laurentpoly.h
index 49c2137..8e9f3a1 100644
--- a/algebra/multivariate_laurentpoly.h
+++ b/algebra/multivariate_laurentpoly.h
@@ -55,6 +55,7 @@ class multivariate_laurent_monomial
 #endif
     return m == e.m;
   }
+  bool operator != (const multivariate_laurent_monomial &e) const { return !operator == (e); }
   
   bool operator == (int x) const
   {
@@ -204,6 +205,12 @@ class multivariate_laurentpoly
       coeffs.push (monomial (VARIABLE, i), c);
   }
   
+  multivariate_laurentpoly (T c, variable, unsigned i, int e)
+  {
+    if (c != 0)
+      coeffs.push (monomial (VARIABLE, i, e), c);
+  }
+  
   multivariate_laurentpoly (T c, const monomial &m)
   {
     if (c != 0)
@@ -212,7 +219,8 @@ class multivariate_laurentpoly
   
   multivariate_laurentpoly (const multivariate_laurentpoly &p) : coeffs(p.coeffs) { }
   multivariate_laurentpoly (copy, const multivariate_laurentpoly &p)
-    : coeffs(COPY2, p.coeffs)
+    // ??? COPY2?
+    : coeffs(COPY, p.coeffs)
   {
   }
 
@@ -243,6 +251,7 @@ class multivariate_laurentpoly
 #endif
     return coeffs == p.coeffs;
   }
+  bool operator != (const multivariate_laurentpoly &p) const { return !operator == (p); }
   
   bool operator == (int x) const
   {
diff --git a/algebra/multivariate_polynomial.h b/algebra/multivariate_polynomial.h
index 75aac48..c35bd32 100644
--- a/algebra/multivariate_polynomial.h
+++ b/algebra/multivariate_polynomial.h
@@ -5,7 +5,7 @@
 template<unsigned n>
 class multivariate_monomial
 {
- private:
+ public:
   unsigned v[n];
   
  public:
@@ -56,6 +56,7 @@ class multivariate_monomial
       }
     return 1;
   }
+  bool operator != (const multivariate_monomial &e) const { return !operator == (e); }
   
   bool operator < (const multivariate_monomial &e) const
   {
@@ -199,7 +200,7 @@ public:
   
   bool is_unit () const;
   
-  bool operator == (multivariate_polynomial p) const
+  bool operator == (const multivariate_polynomial &p) const
   {
 #ifndef NDEBUG
     check ();
@@ -207,6 +208,7 @@ public:
 #endif
     return coeffs == p.coeffs;
   }
+  bool operator != (const multivariate_polynomial &p) const { return !operator == (p); }
   
   bool operator == (int x) const
   {
@@ -263,6 +265,12 @@ public:
   
   monomial common_monomial () const;
   
+  multivariate_polynomial gcd (const multivariate_polynomial &b) const
+  {
+    // ???
+    return multivariate_polynomial (1);
+  }
+  
   pair<multivariate_polynomial, multivariate_polynomial>
     uncommon_factors (multivariate_polynomial b, basedvector<multivariate_polynomial, 1> ds);
   maybe<multivariate_polynomial>
diff --git a/algebra/polynomial.h b/algebra/polynomial.h
index 4aa83c5..4d1771f 100644
--- a/algebra/polynomial.h
+++ b/algebra/polynomial.h
@@ -41,6 +41,7 @@ public:
   
   polynomial (const polynomial &p) : coeffs(p.coeffs) { }
   polynomial (copy, const polynomial &p) : coeffs(COPY, p.coeffs) { }
+  polynomial (reader &r) : coeffs(r) { }
   ~polynomial () { }
   
   polynomial &operator = (const polynomial &p) { coeffs = p.coeffs; return *this; }
@@ -64,7 +65,16 @@ public:
   unsigned degree () const;
   bool is_unit () const;
   
-  bool operator == (polynomial p) const
+  polynomial recip () const
+  {
+    assert (coeffs.card () == 1);
+    
+    pair<unsigned, T> p = coeffs.head ();
+    assert (p.first == 0);
+    return polynomial (p.second.recip ());
+  }
+  
+  bool operator == (const polynomial &p) const
   {
 #ifndef NDEBUG
     check ();
@@ -72,6 +82,7 @@ public:
 #endif
     return coeffs == p.coeffs;
   }
+  bool operator != (const polynomial &p) const { return !operator == (p); }
   
   bool operator == (int x) const
   {
@@ -95,6 +106,7 @@ public:
   polynomial &operator *= (polynomial p) { return operator = (*this * p); }
   polynomial &operator *= (T s);
   
+  // ??? rename
   polynomial &add_term (T c, unsigned e)
   {
     T &c2 = coeffs[e];
@@ -112,27 +124,36 @@ public:
     return r;
   }
   
-  polynomial operator - (polynomial p) const
+  polynomial operator - (const polynomial &p) const
   {
     polynomial r (COPY, *this);
     r -= p;
     return r;
   }
   
-  polynomial operator * (polynomial p) const;
+  polynomial operator * (const polynomial &p) const;
   
-  pair<polynomial, polynomial> divide_with_remainder (polynomial d) const;
+  bool divides (const polynomial &num) const;
   
-  polynomial mod (polynomial d) const;
-  bool divides (polynomial d) const;
-  polynomial divide_exact (polynomial d) const;
+  // *this | num
+  bool operator | (const polynomial &num) const { return divides (num); }
   
-  polynomial gcd (polynomial b) const;
+  tuple<polynomial, polynomial> divide_with_remainder (const polynomial &denom) const;
+  
+  polynomial mod (const polynomial &denom) const;
+
+  polynomial divide_exact (const polynomial &denom) const;
+  polynomial div (const polynomial &denom) const { return divide_exact (denom); }
+  
+  polynomial gcd (const polynomial &b) const;
+  polynomial lcm (const polynomial &b) const;
+  tuple<polynomial, polynomial, polynomial> extended_gcd (const polynomial &b) const;
   
 #ifndef NDEBUG
   void check () const;
 #endif
   
+  void write_self (writer &w) const { write (w, coeffs); }
   static void show_ring () { T::show_ring (); printf ("[x]"); }
   void display_self () const { show_self (); newline (); }
   void show_self () const;
@@ -197,7 +218,7 @@ polynomial<T>::operator *= (T s)
 }
 
 template<class T> polynomial<T>
-polynomial<T>::operator * (polynomial p) const
+polynomial<T>::operator * (const polynomial &p) const
 {
   polynomial r;
   
@@ -213,14 +234,15 @@ polynomial<T>::operator * (polynomial p) const
   return r;
 }
 
-template<class T> pair<polynomial<T>, polynomial<T> > 
-polynomial<T>::divide_with_remainder (polynomial d) const
+template<class T> tuple<polynomial<T>, polynomial<T> > 
+polynomial<T>::divide_with_remainder (const polynomial &d) const
 {
+  // num = *this
   assert (d != 0);
   
   polynomial r (COPY, *this);
   polynomial q;
-
+  
   pair<unsigned, T> d_leading_term = d.coeffs.tail ();
   for (;;)
     {
@@ -240,45 +262,80 @@ polynomial<T>::divide_with_remainder (polynomial d) const
   assert (r == 0 || r.degree () < d.degree ());
   // assert (*this == q*d + r);
   
-  return pair<polynomial, polynomial> (q, r);
+  return make_tuple (q, r);
 }
 
 template<class T> polynomial<T>
-polynomial<T>::mod (polynomial d) const
+polynomial<T>::mod (const polynomial &denom) const
 {
-  pair<polynomial<T>, polynomial<T> > qr = divide_with_remainder (d);
-  return qr.second;
+  polynomial q, r;
+  tie (q, r) = divide_with_remainder (denom);
+  return r;
 }
 
 template<class T> bool
-polynomial<T>::divides (polynomial d) const
+polynomial<T>::divides (const polynomial &num) const
 {
-  pair<polynomial<T>, polynomial<T> > qr = divide_with_remainder (d);
-  return qr.second == 0;
+  // denom = *this
+  polynomial q, r;
+  tie (q, r) = num.divide_with_remainder (*this);
+  return r == 0;
 }
 
 template<class T> polynomial<T>
-polynomial<T>::divide_exact (polynomial d) const
+polynomial<T>::divide_exact (const polynomial &denom) const
 {
-  pair<polynomial<T>, polynomial<T> > qr = divide_with_remainder (d);
-  assert (qr.second == 0);
-  return qr.first;
+  polynomial q, r;
+  tie (q, r) = divide_with_remainder (denom);
+  assert (r == 0);
+  return q;
 }
 
 template<class T> polynomial<T>
-polynomial<T>::gcd (polynomial b) const
+polynomial<T>::gcd (const polynomial &b) const
 {
   polynomial a = *this;
-  
   while (b != 0)
     {
-      pair<polynomial<T>, polynomial<T> > a_qr = a.divide_with_remainder (b);
+      polynomial r;
+      tie (ignore, r) = a.divide_with_remainder (b);
       a = b;
-      b = a_qr.second;
+      b = r;
     }
   return a;
 }
 
+template<class T> polynomial<T>
+polynomial<T>::lcm (const polynomial &b) const
+{
+  // a = *this
+  return divide_exact (gcd (b)) * b;
+}
+
+template<class T> tuple<polynomial<T>, polynomial<T>, polynomial<T> >
+polynomial<T>::extended_gcd (const polynomial &b) const
+{
+  // a = *this
+  if (b == 0)
+    return make_tuple (*this, polynomial (1), polynomial (0));
+  
+  polynomial q, r;
+  tie (q, r) = divide_with_remainder (b);
+  if (r == 0)
+    return make_tuple (b, polynomial (0), polynomial (1));
+  else
+    {
+      polynomial d, s, t;
+      
+      tie (d, s, t) = b.extended_gcd (r);
+      
+      polynomial s2 = t,
+	t2 = s - t*q;
+      
+      return make_tuple (d, s2, t2);
+    }
+}
+
 #ifndef NDEBUG
 template<class T> void
 polynomial<T>::check () const
@@ -338,14 +395,14 @@ class polynomial<Z2>
   
  public:
   polynomial () { }
-  explicit polynomial (int x)
+  polynomial (int x)
   {
     Z2 c (x);
     if (c != 0)
       coeffs.push (0);
   }
   
-  explicit polynomial (Z2 c)
+  polynomial (Z2 c)
   {
     if (c != 0)
       coeffs.push (0);
@@ -380,7 +437,8 @@ class polynomial<Z2>
     return *this;
   }
   
-  bool operator == (polynomial p) const { return coeffs == p.coeffs; }
+  bool operator == (const polynomial &p) const { return coeffs == p.coeffs; }
+  bool operator != (const polynomial &p) const { return !operator == (p); }
   
   bool operator == (int x) const
   {
@@ -395,6 +453,17 @@ class polynomial<Z2>
   bool operator != (int x) const { return !operator == (x); }
   
   unsigned degree () const { return coeffs.tail (); }
+  bool is_unit () const
+  {
+    return coeffs.card () == 1
+      && coeffs.head () == 0;
+  }
+  
+  polynomial recip () const
+  {
+    assert (is_unit ());
+    return 1;
+  }
   
   polynomial &operator += (polynomial p) { coeffs ^= p.coeffs; return *this; }
   polynomial &operator -= (polynomial p) { coeffs ^= p.coeffs; return *this; }
@@ -443,4 +512,3 @@ class polynomial<Z2>
   void display_self () const { show_self (); newline (); }
   void show_self () const;
 };
-
diff --git a/cobordism.cpp b/cobordism.cpp
index 55acf28..72e8200 100644
--- a/cobordism.cpp
+++ b/cobordism.cpp
@@ -19,10 +19,10 @@ public:
   
 public:
   rd_unify ();
-  rd_unify (const rd_unify &); // doesn't exist
+  rd_unify (const rd_unify &) = delete;
   ~rd_unify () { }
   
-  rd_unify &operator = (const rd_unify &); // doesn't exist
+  rd_unify &operator = (const rd_unify &) = delete;
   
   bool unify (const resolution_diagram &rd1, const resolution_diagram &rd2);
   unsigned map_starting_monomial (unsigned m);
diff --git a/cobordism.h b/cobordism.h
index e8dfc47..0365343 100644
--- a/cobordism.h
+++ b/cobordism.h
@@ -160,6 +160,7 @@ class resolution_diagram
   resolution_diagram &operator = (const resolution_diagram &rd);
   
   bool operator == (const resolution_diagram &rd) const;
+  bool operator != (const resolution_diagram &rd) const { return !operator == (rd); }
   
   /* reverse, reverse_orientation leave starting/ending circle numbers
      unchanged. */
@@ -190,10 +191,10 @@ class resolution_diagram_builder
   
  public:
   resolution_diagram_builder ();
-  resolution_diagram_builder (const resolution_diagram_builder &); // doesn't exist
+  resolution_diagram_builder (const resolution_diagram_builder &) = delete;
   ~resolution_diagram_builder () { }
   
-  resolution_diagram_builder &operator = (const resolution_diagram_builder &); // doesn't exist
+  resolution_diagram_builder &operator = (const resolution_diagram_builder &) = delete;
   
   void init (const knot_diagram &d,
 	     smallbitset from_state, const smoothing &from_s,
diff --git a/cube.h b/cube.h
index b3883ac..248995e 100644
--- a/cube.h
+++ b/cube.h
@@ -3,10 +3,10 @@ class map_rules
 {
  public:
   map_rules () { }
-  map_rules (const map_rules &); // doesn't exist
+  map_rules (const map_rules &) = delete;
   virtual ~map_rules () { }
   
-  map_rules &operator = (const map_rules &);  // doesn't exist
+  map_rules &operator = (const map_rules &) = delete;
   
   virtual void map (basedvector<pair<unsigned, unsigned>, 1> &out,
 		    resolution_diagram_builder &rdb) const = 0;
@@ -16,8 +16,8 @@ template<class R>
 class cube /* of resolutions */
 {
 public:
-  typedef typename R::linear_combination linear_combination;
-  typedef typename R::linear_combination_const_iter linear_combination_const_iter;
+  typedef ::linear_combination<R> linear_combination;
+  typedef ::linear_combination_const_iter<R> linear_combination_const_iter;
   
  public:
   bool markedp_only;
@@ -47,6 +47,7 @@ public:
 				    bool reverse_orientation,
 				    unsigned to_reverse) const;
   
+  mod_map<R> compute_dinv (unsigned c);
   mod_map<R> H_i (unsigned c);
   
   mod_map<R> compute_nu () const;
@@ -77,10 +78,10 @@ class twisted_map_rules
 {
  public:
   twisted_map_rules () { }
-  twisted_map_rules (const twisted_map_rules &); // doesn't exist
+  twisted_map_rules (const twisted_map_rules &) = delete;
   virtual ~twisted_map_rules () { }
   
-  map_rules &operator = (const twisted_map_rules &);  // doesn't exist
+  map_rules &operator = (const twisted_map_rules &) = delete;
   
   virtual void map (basedvector<triple<unsigned, unsigned, set<unsigned> >, 1> &out,
 		    resolution_diagram_builder &rdb) const = 0;
@@ -100,10 +101,10 @@ class twisted_cube
     : c(c_)
   { }
   
-  twisted_cube (const twisted_cube &); // doesn't exist
+  twisted_cube (const twisted_cube &) = delete;
   ~twisted_cube () { }
   
-  twisted_cube &operator = (const twisted_cube &); // doesn't exist
+  twisted_cube &operator = (const twisted_cube &) = delete;
   
   mod_map<R> compute_twisted_map (basedvector<int, 1> edge_weight,
 				  unsigned dh,
diff --git a/cube_impl.h b/cube_impl.h
index 87ae5f8..e66ebf1 100644
--- a/cube_impl.h
+++ b/cube_impl.h
@@ -9,7 +9,7 @@ public:
   khC_generators (const cube<R> &c_) : c(c_) { }
   ~khC_generators () { }
   
-  khC_generators &operator = (const khC_generators &); // doesn't exist
+  khC_generators &operator = (const khC_generators &) = delete;
   
   unsigned dim () const { return c.n_generators; }
   unsigned free_rank () const { return c.n_generators; }
@@ -166,7 +166,7 @@ cube<R>::compute_map (unsigned dh, unsigned max_n,
 			continue;
 		      assert (!unsigned_bittest (v_to, p));
 		    }
-
+		  
 		  b[generator (fromstate, v_from)].muladd
 		    (sign, generator (tostate, v_to));
 		}
@@ -187,10 +187,10 @@ class d_rules : public map_rules
 {
 public:
   d_rules () { }
-  d_rules (const d_rules &); // doesn't exist
+  d_rules (const d_rules &) = delete;
   ~d_rules () { }
   
-  d_rules &operator = (const d_rules &); // doesn't exist
+  d_rules &operator = (const d_rules &) = delete;
   
   void map (basedvector<pair<unsigned, unsigned>, 1> &out,
 	    resolution_diagram_builder &rdb) const
@@ -214,10 +214,10 @@ class twin_arrows_P_rules : public map_rules
 {
 public:
   twin_arrows_P_rules () { }
-  twin_arrows_P_rules (const twin_arrows_P_rules &); // doesn't exist
+  twin_arrows_P_rules (const twin_arrows_P_rules &) = delete;
   ~twin_arrows_P_rules () { }
   
-  twin_arrows_P_rules &operator = (const twin_arrows_P_rules &); // doesn't exist
+  twin_arrows_P_rules &operator = (const twin_arrows_P_rules &) = delete;
   
   void map (basedvector<pair<unsigned, unsigned>, 1> &out,
 	    resolution_diagram_builder &rdb) const
@@ -302,7 +302,7 @@ cube<R>::compute_X (unsigned p) const
 template<class R> mod_map<R>
 cube<R>::H_i (unsigned c)
 {
-  mod_map<R> b (khC, 0);
+  map_builder<R> b (khC, 0);
   for (unsigned i = 0; i < n_resolutions; i ++)
     {
       if (unsigned_bittest (i, c))
@@ -319,8 +319,23 @@ cube<R>::H_i (unsigned c)
       
       unsigned from = from_s.crossing_from_circle (kd, c),
 	to = from_s.crossing_to_circle (kd, c);
+      
+      int sign = 1;
+      for (unsigned j = 1; j < c; j ++)
+        {
+          if (unsigned_bittest (i, j))
+            sign *= -1;
+        }
+      
       for (unsigned j = 0; j < from_s.num_monomials (); j ++)
 	{
+	  if (markedp_only)
+	    {
+	      unsigned p = from_s.edge_circle[kd.marked_edge];
+	      if (unsigned_bittest (j, p))
+		continue;
+	    }
+	  
 	  unsigned j2 = 0;
 	  for (unsigned_const_iter k = j; k; k ++)
 	    {
@@ -336,7 +351,7 @@ cube<R>::H_i (unsigned c)
 				    to_s.crossing_from_circle (kd, c));
 	      j2 = unsigned_bitset (j2,
 				    to_s.crossing_to_circle (kd, c));
-	      v.muladd (1, generator (i2, j2));
+	      v.muladd (-sign, generator (i2, j2));
 	    }
 	  else if (from != to
 		   && !unsigned_bittest (j, from)
@@ -344,7 +359,7 @@ cube<R>::H_i (unsigned c)
 	    {
 	      assert (! unsigned_bittest (j2,
 					  to_s.crossing_from_circle (kd, c)));
-	      v.muladd (1, generator (i2, j2));
+	      v.muladd (sign, generator (i2, j2));
 	    }
 	}
     }
@@ -356,6 +371,110 @@ cube<R>::H_i (unsigned c)
   return H_c;
 }
 
+template<class R> mod_map<R>
+cube<R>::compute_dinv (unsigned c)
+{
+  map_builder<R> p (khC);
+  for (unsigned i = 0; i < n_resolutions; i ++)
+    {
+      if (!unsigned_bittest (i, c))
+	continue;
+      
+      int sign = 1;
+      for (unsigned j = 1; j < c; j ++)
+	{
+	  if (unsigned_bittest (i, j))
+	    sign *= -1;
+	}
+      
+      smoothing from_s (kd, smallbitset (n_crossings, i));
+      
+      unsigned i2 = unsigned_bitclear (i, c);
+      smoothing to_s (kd, smallbitset (n_crossings, i2));
+      
+      basedvector<unsigned, 1> from_circle_edge_rep (from_s.n_circles);
+      for (unsigned j = 1; j <= kd.num_edges (); j ++)
+	from_circle_edge_rep[from_s.edge_circle[j]] = j;
+      
+      unsigned a = from_s.crossing_from_circle (kd, c),
+	b = from_s.crossing_to_circle (kd, c);
+      unsigned x = to_s.crossing_from_circle (kd, c),
+	y = to_s.crossing_to_circle (kd, c);
+      for (unsigned j = 0; j < from_s.num_monomials (); j ++)
+	{
+	  if (markedp_only)
+	    {
+	      unsigned p = from_s.edge_circle[kd.marked_edge];
+	      if (unsigned_bittest (j, p))
+		continue;
+	    }
+	  
+	  unsigned j2 = 0;
+	  for (unsigned_const_iter k = j; k; k ++)
+	    {
+	      unsigned s = k.val ();
+	      j2 = unsigned_bitset (j2, to_s.edge_circle[from_circle_edge_rep[s]]);
+	    }
+	  
+	  if (a == b)
+	    {
+	      // split
+	      assert (x != y);
+	      
+	      if (unsigned_bittest (j, a))
+		{
+		  // 1 -> x + y
+		  j2 = unsigned_bitset (j2, x);
+		  j2 = unsigned_bitclear (j2, y);
+		  
+		  p[generator (i, j)].muladd (sign, generator (i2, j2));
+		  
+		  j2 = unsigned_bitclear (j2, x);
+		  j2 = unsigned_bitset (j2, y);
+		  
+		  p[generator (i, j)].muladd (sign, generator (i2, j2));
+		}
+	      else
+		{
+		  // a -> xy
+		  j2 = unsigned_bitclear (j2, x);
+		  j2 = unsigned_bitclear (j2, y);
+		  
+		  p[generator (i, j)].muladd (sign, generator (i2, j2));
+		}
+	    }
+	  else
+	    {
+	      // join
+	      assert (x == y);
+	      
+	      if (unsigned_bittest (j, a)
+		  && unsigned_bittest (j, b))
+		{
+		  // 1 -> 1
+		  j2 = unsigned_bitset (j2, x);
+		  
+		  p[generator (i, j)].muladd (sign, generator (i2, j2));
+		}
+	      else if ((unsigned_bittest (j, a)
+			&& !unsigned_bittest (j, b))
+		       || (!unsigned_bittest (j, a)
+			   && unsigned_bittest (j, b)))
+		{
+		  // a, b -> x
+		  j2 = unsigned_bitclear (j2, x);
+		  
+		  p[generator (i, j)].muladd (sign, generator (i2, j2));
+		}
+	    }
+	}
+    }
+  
+  mod_map<R> dinv (p);
+  dinv.check_grading (grading (-1, -2));
+  return dinv;
+}
+
 template<class R> void
 cube<R>::check_reverse_crossings ()
 {
@@ -415,6 +534,50 @@ cube<R>::cube (knot_diagram &kd_, bool markedp_only_)
     {
       smallbitset state (n_crossings, i);
       s.init (kd, state);
+      
+#if 1
+      unsigned fromstate = i;
+      smoothing &from_s = s;
+      
+      unsigned n_zerocrossings = n_crossings - unsigned_bitcount (fromstate);
+      unsigned n_cobordisms = ((unsigned)1) << n_zerocrossings;
+      for (unsigned j = 0; j < n_cobordisms; j ++)
+	{
+	  unsigned tostate = unsigned_pack (n_crossings, fromstate, j);
+	  unsigned crossings = tostate & ~fromstate;
+	  
+	  smoothing to_s (kd, smallbitset (n_crossings, tostate));
+	  
+	  set<unsigned> starting_circles,
+	    ending_circles;
+	  for (unsigned_const_iter k = crossings; k; k ++)
+	    {
+	      unsigned c = k.val ();
+	      
+	      unsigned starting_from = s.ept_circle (kd, kd.crossings[c][2]),
+		starting_to = s.ept_circle (kd, kd.crossings[c][4]);
+	      starting_circles += starting_from;
+	      starting_circles += starting_to;
+	      
+	      unsigned ending_from = to_s.ept_circle (kd, kd.crossings[c][2]),
+		ending_to = to_s.ept_circle (kd, kd.crossings[c][4]);
+	      ending_circles += ending_from;
+	      ending_circles += ending_to;
+	    }
+	  if (starting_circles.card () == 1
+	      && ending_circles.card () == 1)
+	    {
+	      unsigned k = unsigned_bitcount (crossings);
+	      
+#if 0
+	      s.show_self (kd, state);
+	      printf (" crossings ");  show (smallbitset (n_crossings, crossings));
+	      newline ();
+#endif
+	    }
+	}
+#endif
+      
       resolution_circles[i] = s.n_circles;
       resolution_generator1[i] = n_generators + 1;
       n_generators += s.num_generators (markedp_only);
@@ -676,10 +839,10 @@ class twisted_barE_rules : public twisted_map_rules
 {
 public:
   twisted_barE_rules () { }
-  twisted_barE_rules (const twisted_barE_rules &); // doesn't exist
+  twisted_barE_rules (const twisted_barE_rules &) = delete;
   ~twisted_barE_rules () { }
   
-  twisted_barE_rules &operator = (const twisted_barE_rules &); // doesn't exist
+  twisted_barE_rules &operator = (const twisted_barE_rules &) = delete;
   
   void map (basedvector<triple<unsigned, unsigned, set<unsigned> >, 1> &out,
 	    resolution_diagram_builder &rdb) const
diff --git a/gss.cpp b/gss.cpp
deleted file mode 100644
index 3f8f28d..0000000
--- a/gss.cpp
+++ /dev/null
@@ -1,126 +0,0 @@
-
-#include <knotkit.h>
-
-const char *program_name;
-
-void
-usage ()
-{
-  printf ("usage: %s [-f] [-h] [-o <file>] [-v] <knot>\n", program_name);
-  printf ("  compute Szabo's geometric spectral sequence for knot <knot>\n");
-  printf ("options:\n");
-  printf ("  -f        : compute unreduced spectral sequence\n");
-  printf ("                (reduced is the default)\n");
-  printf ("  -h        : print this message\n");
-  printf ("  -o <file> : write output to <file>\n");
-  printf ("                (stdout is the default)\n");
-  printf ("  -v        : verbose: report progress as the computation proceeds\n");
-  printf ("<knot> can be one of:\n");
-  printf ("  - the unknot, e.g. U or unknot\n");
-  printf ("  - a torus knot, e.g. T(2,3)\n");
-  printf ("  - a Rolfsen table knot, e.g. 10_124\n");
-  printf ("  - a Hoste-Thistlethwaite-Weeks knot, e.g. 11a12 or 12n214\n");
-  printf ("  - a Morwen Thistlethwaite link, e.g. L9a21 or L14n7631\n");
-  printf ("  - a planar diagram, e.g.\n");
-  printf ("      PD[X[1, 4, 2, 5], X[3, 6, 4, 1], X[5, 2, 6, 3]] or\n");
-  printf ("      PD[[1, 4, 2, 5], [3, 6, 4, 1], [5, 2, 6, 3]]\n");
-  printf ("  - a Dowker-Thistlethwaite code, e.g.\n");
-  printf ("      DTCode[6,8,2,4],\n");
-  printf ("      DT[dadbcda] or\n");
-  printf ("      DT[{6, -8}, {-10, 12, -14, 2, -4}]\n");
-  printf ("  - a braid, e.g. BR[2, {-1, -1, -1}]\n");
-}
-
-int
-main (int argc, char **argv)
-{
-  program_name = argv[0];
-  
-  bool reduced = 1;
-  const char *knot = 0;
-  const char *file = 0;
-  
-  for (int i = 1; i < argc; i ++)
-    {
-      if (argv[i][0] == '-')
-	{
-	  if (strcmp (argv[i], "-f") == 0)
-	    reduced = 0;
-	  else if (strcmp (argv[i], "-h") == 0)
-	    {
-	      usage ();
-	      exit (EXIT_SUCCESS);
-	    }
-	  else if (!strcmp (argv[i], "-v"))
-	    verbose = 1;
-	  else if (!strcmp (argv[i], "-o"))
-	    {
-	      i ++;
-	      if (i == argc)
-		{
-		  fprintf (stderr, "error: missing argument to option `-o'\n");
-		  exit (EXIT_FAILURE);
-		}
-	      file = argv[i];
-	    }
-	  else
-	    {
-	      fprintf (stderr, "error: unknown argument `%s'\n", argv[1]);
-	      usage ();
-	      exit (EXIT_FAILURE);
-	    }
-	}
-      else
-	{
-	  if (knot)
-	    {
-	      fprintf (stderr, "error: too many arguments\n");
-	      usage ();
-	      exit (EXIT_FAILURE);
-	    }
-	  else
-	    knot = argv[i];
-	}
-    }
-  
-  if (!knot)
-    {
-      fprintf (stderr, "error: missing <knot> argument\n");
-      usage ();
-      exit (EXIT_FAILURE);
-    }
-  
-  FILE *outfp = 0;
-  if (file)
-    {
-      outfp = fopen (file, "w");
-      if (!outfp)
-	{
-	  stderror ("fopen: %s", file);
-	  exit (EXIT_FAILURE);
-	}
-    }
-  else
-    outfp = stdout;
-  
-  knot_diagram d = parse_knot (knot);
-  d.marked_edge = 1;
-  cube<Z2> c (d, reduced);
-  
-  fprintf (outfp, "\\documentclass{article}\n\
-\\usepackage{amsmath, tikz, hyperref}\n\
-\\DeclareMathOperator{\\rank}{rank}\n\
-\\setlength{\\parindent}{0pt}\n\
-\n\
-\\begin{document}\n\
-\\pagestyle{empty}\n\
-\\tableofcontents\n\
-\\sloppy\n");
-  
-  compute_szabo_sseq (c).texshow (outfp, d.name);
-  
-  fprintf (outfp, "\\end{document}\n");
-  
-  if (file)
-    fclose (outfp);
-}
diff --git a/kk.cpp b/kk.cpp
new file mode 100644
index 0000000..0d45025
--- /dev/null
+++ b/kk.cpp
@@ -0,0 +1,666 @@
+
+#include <knotkit.h>
+
+const char *program_name;
+
+void
+usage ()
+{
+  printf ("usage: %s <invariant> [options...] <link>\n", program_name);
+  printf ("  compute <invariant> for knot or link <link>\n");
+  printf ("<invariant> can be one of:\n");
+  printf ("  kh: Khovanov homology\n");
+  printf ("  gss: Szabo's geometric spectral sequence\n");
+  printf ("  lsss: Batson-Seed link splitting spectral sequence\n");
+  printf ("    component weights are 0, 1, ..., m\n");
+  printf ("  sq2: Lipshitz-Sarkar Steenrod square on Z/2 Kh\n");
+  printf ("    output suitable for Sage\n");
+  printf ("  leess: spectral sequence coming from Bar-Natan analogue of Lee's\n");
+  printf ("    deformation of Khovanov's complex (whew!)\n");
+  printf ("  s: Rasmussen's s-invariant coming from lee\n");
+  printf ("output:\n");
+  printf ("    kh, gss, lsss, leess: .tex file\n");
+  printf ("    sq2: text in Sage format\n");
+  printf ("    s: text\n");
+  printf ("options:\n");
+  printf ("  -r         : compute reduced theory\n");
+  printf ("  -h         : print this message\n");
+  printf ("  -o <file>  : write output to <file>\n");
+  printf ("                (stdout is the default)\n");
+  printf ("  -f <field> : ground field (if applicable)\n");
+  printf ("                (Z2 is the default)\n");
+  printf ("  -v         : verbose: report progress as the computation proceeds\n");
+  printf ("<field> can be one of:\n");
+  printf ("  Z2, Z3, Q\n");
+  printf ("<link> can be one of:\n");
+  printf ("  - the unknot, e.g. U or unknot\n");
+  printf ("  - a torus knot, e.g. T(2,3)\n");
+  printf ("  - a Rolfsen table knot, e.g. 10_124\n");
+  printf ("  - a Hoste-Thistlethwaite-Weeks knot, e.g. 11a12 or 12n214\n");
+  printf ("  - a Morwen Thistlethwaite link, e.g. L8n9 or L13n8862\n");
+  printf ("  - a planar diagram, e.g.\n");
+  printf ("      PD[X[1, 4, 2, 5], X[3, 6, 4, 1], X[5, 2, 6, 3]] or\n");
+  printf ("      PD[[1, 4, 2, 5], [3, 6, 4, 1], [5, 2, 6, 3]]\n");
+  printf ("  - a Dowker-Thistlethwaite code, e.g.\n");
+  printf ("      DTCode[6,8,2,4],\n");
+  printf ("      DT[dadbcda] or\n");
+  printf ("      DT[{6, -8}, {-10, 12, -14, 2, -4}]\n");
+  printf ("  - a braid, e.g. BR[2, {-1, -1, -1}]\n");
+  printf ("  - disjoint union (juxtaposition), e.g. T(2,3) U\n");
+}
+
+FILE *outfp = stdout;
+
+void tex_header ()
+{
+  fprintf (outfp, "\\documentclass{article}\n\
+\\usepackage{amsmath, tikz, hyperref}\n\
+\\DeclareMathOperator{\\rank}{rank}\n\
+\\setlength{\\parindent}{0pt}\n\
+\n\
+\\begin{document}\n\
+\\pagestyle{empty}\n\
+\\sloppy\n");
+}
+
+void tex_footer ()
+{
+  fprintf (outfp, "\\end{document}\n");
+}
+
+const char *knot = 0;
+const char *invariant = 0;
+const char *field = "Z2";
+
+knot_diagram kd;
+
+bool reduced = 0;
+
+class hg_grading_mapper
+{
+  unsigned m;
+  
+public:
+  hg_grading_mapper (unsigned m_) : m(m_) { }
+  
+  grading operator () (grading hq) const
+  {
+    int t = hq.q - (int)m;
+    if (reduced)
+      t --;
+    
+    assert (is_even (t));
+    return grading (hq.h, t / 2);
+  }
+  
+  grading map_delta (grading d_hq) const
+  {
+    assert (is_even (d_hq.q));
+    return grading (d_hq.h, d_hq.q / 2);
+  }
+  
+  void x_label (FILE *fp, int h) const
+  {
+    fprintf (fp, "%d", h);
+  }
+  void y_label (FILE *fp, int g) const
+  {
+    unsigned q = 2*g + (int)m;
+    if (reduced)
+      q ++;
+    
+    fprintf (fp, "%d", q);
+  }
+};
+
+template<class R> mod_map<R>
+compute_link_splitting_d (knot_diagram &kd,
+			  cube<R> &c,
+			  basedvector<R, 1> comp_weight)
+{
+  unsigned n = kd.num_components ();
+  
+  unionfind<1> u (kd.num_edges ());
+  for (unsigned i = 1; i <= kd.n_crossings; i ++)
+    {
+      u.join (kd.ept_edge (kd.crossings[i][1]),
+	      kd.ept_edge (kd.crossings[i][3]));
+      u.join (kd.ept_edge (kd.crossings[i][2]),
+	      kd.ept_edge (kd.crossings[i][4]));
+    }
+	
+  map<unsigned, unsigned> root_comp;
+  unsigned t = 0;
+  for (unsigned i = 1; i <= kd.num_edges (); i ++)
+    {
+      if (u.find (i) == i)
+	{
+	  ++ t;
+	  root_comp.push (i, t);
+	}
+    }
+  assert (t == n);
+  
+  assert (comp_weight.size () == n);
+  
+  map<unsigned, R> crossing_over_sign;
+	
+  // crossings
+  set<unsigned> pending;
+  set<unsigned> finished;
+	
+  crossing_over_sign.push (1, R (1));
+  pending.push (1);
+  
+  while (pending.card () > 0)
+    {
+      unsigned x = pending.pop ();
+      finished.push (x);
+      
+      R s = crossing_over_sign(x);
+      
+      for (unsigned j = 1; j <= 4; j ++)
+	{
+	  unsigned p = kd.crossings[x][j];
+	  R t = kd.is_over_ept (p) ? s : -s;  // sign of (x, p)
+	  
+	  unsigned q = kd.edge_other_ept (p);
+	  unsigned x2 = kd.ept_crossing[q];
+	  
+	  R u = kd.is_over_ept (q) ? -t : t;
+	  
+	  if (crossing_over_sign % x2)
+	    assert (crossing_over_sign(x2) == u);
+	  else
+	    crossing_over_sign.push (x2, u);
+	  
+	  if (! (finished % x2))
+	    pending += x2;
+	}
+    }
+  assert (finished.card () == kd.n_crossings);
+  
+  mod_map<R> untwisted_d = c.compute_d (1, 0, 0, 0, 0);
+  assert (untwisted_d.compose (untwisted_d) == 0);
+  
+  mod_map<R> d = untwisted_d;
+  for (unsigned x = 1; x <= kd.n_crossings; x ++)
+    {
+      unsigned p1 = kd.crossings[x][1],
+	p2 = kd.crossings[x][2];
+      assert (kd.is_over_ept (p2));
+      
+      unsigned r1 = u.find (kd.ept_edge (p1)),
+	r2 = u.find (kd.ept_edge (p2));
+      
+      unsigned c1 = root_comp(r1),
+	c2 = root_comp(r2);
+      
+      if (c1 != c2)
+	{
+	  R s = crossing_over_sign(x);
+		
+	  R w_under = comp_weight[c1];
+	  R w_over = comp_weight[c2];
+		
+	  d = d + c.compute_dinv (x)*(s*(w_over - w_under));
+	}
+    }
+  
+  assert (d.compose (d) == 0);
+  return d;
+}
+
+void
+compute_gss ()
+{
+  cube<Z2> c (kd, reduced);
+  ptr<const module<Z2> > C = c.khC;
+  mod_map<Z2> d = c.compute_d (0, 0, 0, 0, 0);
+  
+  unsigned m = kd.num_components ();
+  hg_grading_mapper mapper (m);
+  
+  sseq_bounds b (C, mapper);
+  basedvector<sseq_page, 1> pages;
+  
+  unsigned k = 1;
+  for (;;)
+    {
+      chain_complex_simplifier<Z2> s (C, d,
+				      maybe<int> (k), maybe<int> (2*k - 2));
+      C = s.new_C;
+      d = s.new_d;
+      k ++;
+      
+      grading dk_gr (k, 2*k - 2);
+      pages.append (sseq_page (b, k, dk_gr, d.graded_piece (dk_gr), mapper));
+      if (d == 0)
+	break;
+    }
+  
+  sseq ss (b, pages);
+  
+  tex_header ();
+  fprintf (outfp, "$E_k = %s^{Sz}_k(\\verb~%s~; \\verb~%s~)$:\\\\\n",
+	       (reduced
+		? "\\widetilde{E}"
+		: "E"),
+	   knot, field);
+  ss.texshow (outfp, mapper);
+  tex_footer ();
+}
+
+template<class R> void
+compute_invariant ()
+{
+  if (!strcmp (invariant, "kh"))
+    {
+      cube<R> c (kd, reduced);
+      ptr<const module<R> > C = c.khC;
+      mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
+      
+      unsigned m = kd.num_components ();
+      hg_grading_mapper mapper (m);
+      
+      chain_complex_simplifier<R> s (C, d,
+				     maybe<int> (1), maybe<int> (0));
+      C = s.new_C;
+      d = s.new_d;
+      
+      sseq_bounds b (C, mapper);
+      sseq_page pg (b, 2, grading (0, 0), mod_map<R> (C), mapper);
+      
+      tex_header ();
+      
+      fprintf (outfp, "Kh = $%s(\\verb~%s~; \\verb~%s~)$:\\\\\n",
+	       (reduced
+		? "\\widetilde{Kh}"
+		: "Kh"),
+	       knot,
+	       field);
+      fprintf (outfp, "$\\rank Kh = %d$\\\\\n", C->dim ());
+      
+      char buf[1000];
+      sprintf (buf, "$Kh$");
+      pg.texshow (outfp, b, buf, mapper);
+      
+      tex_footer ();
+    }
+  else if (!strcmp (invariant, "lsss"))
+    {
+      cube<R> c (kd, reduced);
+      ptr<const module<R> > C = c.khC;
+      
+      unsigned m = kd.num_components ();
+      basedvector<R, 1> comp_weight (m);
+      for (unsigned i = 1; i <= m; i ++)
+	comp_weight[i] = R ((int)(i - 1));
+      
+      mod_map<R> d = compute_link_splitting_d (kd, c, comp_weight);
+      
+      hg_grading_mapper mapper (m);
+      
+      sseq_bounds b (C, mapper);
+      basedvector<sseq_page, 1> pages;
+      
+      int k = 0;
+      for (;;)
+	{
+	  chain_complex_simplifier<R> s (C, d,
+					 maybe<int> (1 - 2*k), maybe<int> (-2*k));
+	  C = s.new_C;
+	  d = s.new_d;
+	  k ++;
+	  
+	  grading dk_gr (1 - 2*k, -2*k);
+	  pages.append (sseq_page (b, k, dk_gr, d.graded_piece (dk_gr), mapper));
+	  if (d == 0)
+	    break;
+	}
+      
+      sseq ss (b, pages);
+      
+      tex_header ();
+      fprintf (outfp, "$E_k = %s^{BS}_k({}^{%d}\\verb~%s~; \\verb~%s~)$:\\\\\n",
+	       (reduced
+		? "\\widetilde{E}"
+		: "E"),
+	       m,
+	       knot,
+	       field);
+      ss.texshow (outfp, mapper);
+      tex_footer ();
+    }
+  else if (!strcmp (invariant, "leess"))
+    {
+      cube<R> c (kd, reduced);
+      ptr<const module<R> > C = c.khC;
+      
+      mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
+      for (unsigned i = 1; i <= kd.n_crossings; i ++)
+	d = d + c.H_i (i);
+      assert (d.compose (d) == 0);
+      
+      unsigned m = kd.num_components ();
+      hg_grading_mapper mapper (m);
+      
+      sseq_bounds b (C, mapper);
+      basedvector<sseq_page, 1> pages;
+      
+      int k = 0;
+      for (;;)
+	{
+	  chain_complex_simplifier<R> s (C, d,
+					 maybe<int> (1), maybe<int> (2*k));
+	  C = s.new_C;
+	  d = s.new_d;
+	  k ++;
+	  
+	  grading dk_gr (1, 2*k);
+	  pages.append (sseq_page (b, k, dk_gr, d.graded_piece (dk_gr), mapper));
+	  if (d == 0)
+	    break;
+	}
+      
+      sseq ss (b, pages);
+      
+      tex_header ();
+      fprintf (outfp, "$E_k = %s^{BN}_k(\\verb~%s~; \\verb~%s~)$:\\\\\n",
+	       (reduced
+		? "\\widetilde{E}"
+		: "E"),
+	       knot, field);
+      ss.texshow (outfp, mapper);
+      tex_footer ();
+    }
+  else if (!strcmp (invariant, "s"))
+    {
+      unsigned m = kd.num_components ();
+      if (m != 1)
+	{
+	  fprintf (stderr, "error: s-invariant only defined for knots\n");
+	  exit (EXIT_FAILURE);
+	}
+      
+      cube<R> c (kd, 0);
+      ptr<const module<R> > C = c.khC;
+      
+      mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
+      for (unsigned i = 1; i <= kd.n_crossings; i ++)
+	d = d + c.H_i (i);
+      assert (d.compose (d) == 0);
+      
+      int k = 0;
+      for (;;)
+	{
+	  chain_complex_simplifier<R> s (C, d,
+					 maybe<int> (1), maybe<int> (2*k));
+	  C = s.new_C;
+	  d = s.new_d;
+	  k ++;
+	  
+	  if (d == 0)
+	    break;
+	}
+      
+      assert (C->dim () == 2);
+      grading gr1 = C->generator_grading (1),
+	gr2 = C->generator_grading (2);
+      
+      int qmin = gr1.q,
+	qmax = gr2.q;
+      if (qmax < qmin)
+	std::swap (qmin, qmax);
+      
+      assert (qmax == qmin + 2);
+      
+      fprintf (outfp, "s(%s; %s) = %d\n", knot, field, qmin + 1);
+    }
+  else 
+    {
+      fprintf (stderr, "error: unknown invariant %s\n", invariant);
+      exit (EXIT_FAILURE);
+    }
+}
+
+void
+sage_show (FILE *fp,
+	   map<grading, basedvector<unsigned, 1> > hq_gens,
+	   ptr<const module<Z2> > H)
+{
+  fprintf (fp, "hom={");
+  bool first = 1;
+  for (map<grading, basedvector<unsigned, 1> >::const_iter i = hq_gens; i; i ++)
+    {
+      if (first)
+	first = 0;
+      else
+	fprintf (fp, ", ");
+      fprintf (fp, "(%d, %d): %d", i.key ().h, i.key ().q, i.val ().size ());
+    }      
+  fprintf (fp, "}\n");
+}
+
+void
+sage_show (FILE *fp,
+	   map<grading, basedvector<unsigned, 1> > hq_gens,
+	   std::string name,
+	   grading delta,
+	   mod_map<Z2> f)
+{
+  fprintf (fp, "%s={", name.c_str ());
+  
+  bool first = 1;
+  for (map<grading, basedvector<unsigned, 1> >::const_iter i = hq_gens; i; i ++)
+    {
+      grading from_hq = i.key ();
+      basedvector<unsigned, 1> from_gens = i.val ();
+      
+      grading to_hq = from_hq + delta;
+      if (hq_gens % to_hq)
+	{
+	  basedvector<unsigned, 1> to_gens = hq_gens(to_hq);
+	  
+	  if (first)
+	    first = 0;
+	  else
+	    fprintf (fp, ", ");
+	  fprintf (fp, "(%d, %d): [", from_hq.h, from_hq.q);
+	  bool first2 = 1;
+	  for (unsigned j = 1; j <= from_gens.size (); j ++)
+	    {
+	      unsigned gj = from_gens[j];
+	      
+	      if (first2)
+		first2 = 0;
+	      else
+		fprintf (fp, ", ");
+	      fprintf (fp, "(");
+	      for (unsigned k = 1; k <= to_gens.size (); k ++)
+		{
+		  unsigned gk = to_gens[k];
+		  
+		  if (k > 1)
+		    fprintf (fp, ", ");
+		  if (f[gj](gk) == 1)
+		    fprintf (fp, "1");
+		  else
+		    {
+		      assert (f[gj](gk) == 0);
+		      fprintf (fp, "0");
+		    }
+		}
+	      fprintf (fp, ")");
+	    }
+	  fprintf (fp, "]");
+	}
+    }
+  fprintf (fp, "}\n");
+}
+
+void
+sage_show_khsq (FILE *fp,
+		ptr<const module<Z2> > H,
+		mod_map<Z2> sq1,
+		mod_map<Z2> sq2)
+{
+  unsigned n = H->dim ();
+  
+  map<grading, basedvector<unsigned, 1> > hq_gens;
+  for (unsigned i = 1; i <= H->dim (); i ++)
+    hq_gens[H->generator_grading (i)].append (i);
+  
+  unsigned t = 0;
+  for (map<grading, basedvector<unsigned, 1> >::const_iter i = hq_gens; i; i ++)
+    t += i.val ().size ();
+  assert (t == n);
+  
+  sage_show (fp, hq_gens, H);
+  sage_show (fp, hq_gens, "sq1", grading (1, 0), sq1);
+  sage_show (fp, hq_gens, "sq2", grading (2, 0), sq2);
+}
+
+void
+compute_sq2 ()
+{
+  cube<Z2> c (kd);
+  mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
+  
+  chain_complex_simplifier<Z2> s (c.khC, d, maybe<int> (1), maybe<int> (0));
+  assert (s.new_d == 0);
+  
+  steenrod_square sq (c, d, s);
+  mod_map<Z2> sq1 = sq.sq1 ();
+  mod_map<Z2> sq2 = sq.sq2 ();
+  
+  ptr<const module<Z2> > H = sq1.domain ();
+  
+  sage_show_khsq (outfp, H, sq1, sq2);
+}
+
+int
+main (int argc, char **argv)
+{
+  program_name = argv[0];
+  
+  const char *file = 0;
+  
+  for (int i = 1; i < argc; i ++)
+    {
+      if (argv[i][0] == '-')
+	{
+	  if (strcmp (argv[i], "-r") == 0)
+	    reduced = 1;
+	  else if (strcmp (argv[i], "-h") == 0)
+	    {
+	      usage ();
+	      exit (EXIT_SUCCESS);
+	    }
+	  else if (!strcmp (argv[i], "-v"))
+	    verbose = 1;
+	  else if (!strcmp (argv[i], "-f"))
+	    {
+	      i ++;
+	      if (i == argc)
+		{
+		  fprintf (stderr, "error: missing argument to option `-f'\n");
+		  exit (EXIT_FAILURE);
+		}
+	      field = argv[i];
+	    }
+	  else if (!strcmp (argv[i], "-o"))
+	    {
+	      i ++;
+	      if (i == argc)
+		{
+		  fprintf (stderr, "error: missing argument to option `-o'\n");
+		  exit (EXIT_FAILURE);
+		}
+	      file = argv[i];
+	    }
+	  else
+	    {
+	      fprintf (stderr, "error: unknown argument `%s'\n", argv[1]);
+	      fprintf (stderr, "  use -h for usage\n");
+	      exit (EXIT_FAILURE);
+	    }
+	}
+      else
+	{
+	  if (knot)
+	    {
+	      fprintf (stderr, "error: too many arguments\n");
+	      fprintf (stderr, "  use -h for usage\n");
+	      exit (EXIT_FAILURE);
+	    }
+	  else if (invariant)
+	    knot = argv[i];
+	  else
+	    {
+	      assert (invariant == 0);
+	      invariant = argv[i];
+	    }
+	}
+    }
+  
+  if (!knot)
+    {
+      fprintf (stderr, "error: too few arguments, <invariant> or <knot> missing\n");
+      fprintf (stderr, "  use -h for usage\n");
+      exit (EXIT_FAILURE);
+    }
+  
+  if (file)
+    {
+      outfp = fopen (file, "w");
+      if (!outfp)
+	{
+	  stderror ("fopen: %s", file);
+	  exit (EXIT_FAILURE);
+	}
+    }
+  else
+    outfp = stdout;
+  
+  kd = parse_knot (knot);
+  kd.marked_edge = 1;
+  
+  if (!strcmp (invariant, "sq2"))
+    {
+      if (strcmp (field, "Z2"))
+	{
+	  fprintf (stderr, "warning: sq2 only defined over Z2, ignoring -f %s\n", field);
+	  field = "Z2";
+	}
+      
+      compute_sq2 ();
+    }
+  else if (!strcmp (invariant, "gss"))
+    {
+      if (strcmp (field, "Z2"))
+	{
+	  fprintf (stderr, "warning: gss only defined over Z2, ignoring -f %s\n", field);
+	  field = "Z2";
+	}
+      
+      compute_gss ();
+    }
+  else
+    {
+      if (!strcmp (field, "Z2"))
+	compute_invariant<Z2> ();
+      else if (!strcmp (field, "Z3"))
+	compute_invariant<Zp<3> > ();
+      else if (!strcmp (field, "Q"))
+	compute_invariant<Q> ();
+      else
+	{
+	  fprintf (stderr, "error: unknown field %s\n", field);
+	  exit (EXIT_FAILURE);
+	}
+    }
+  
+  if (file)
+    fclose (outfp);
+}
diff --git a/knot_diagram.cpp b/knot_diagram.cpp
index 5b5e803..ea6c1b2 100644
--- a/knot_diagram.cpp
+++ b/knot_diagram.cpp
@@ -22,7 +22,7 @@ knot_diagram::knot_diagram (const planar_diagram &pd)
       crossings[c] = basedvector<unsigned, 1> (4);
     }
 
-  smallbitset done (num_edges ());
+  set<unsigned> done;
   for (unsigned c0 = 1; c0 <= n_crossings; c0 ++)
     {
       for (unsigned i0 = 1; i0 <= 2; i0 ++)
@@ -82,10 +82,10 @@ public:
   
 public:
   dt_layout (const dt_code &dt, knot_diagram &kd_);
-  dt_layout (const dt_layout &); //  doesn't exist
+  dt_layout (const dt_layout &) = delete;
   ~dt_layout () { }
 
-  dt_layout &operator = (const dt_layout &); // doesn't exist
+  dt_layout &operator = (const dt_layout &) = delete;
   
   unsigned find_crossing (unsigned prevc, unsigned previ, unsigned target,
 			  bool under, unsigned dir);
@@ -454,6 +454,60 @@ knot_diagram::knot_diagram (sublink,
   calculate_nminus_nplus ();
 }
 
+knot_diagram::knot_diagram (disjoint_union,
+			    const knot_diagram &kd1,
+			    const knot_diagram &kd2)
+  : name(kd1.name + "+" + kd2.name),
+    n_crossings(kd1.n_crossings + kd2.n_crossings),
+    marked_edge(0),
+    crossings(n_crossings),
+    nminus(kd1.nminus + kd2.nminus),
+    nplus(kd1.nplus + kd2.nplus)
+{
+  assert (kd1.marked_edge == 0);
+  assert (kd2.marked_edge == 0);
+  
+  for (unsigned i = 1; i <= n_crossings; i ++)
+    crossings[i] = basedvector<unsigned, 1> (4);
+  
+  for (unsigned i = 1; i <= kd1.n_crossings; i ++)
+    for (unsigned j = 1; j <= 4; j ++)
+      crossings[i][j] = kd1.crossings[i][j];
+  
+  for (unsigned e = 1; e <= kd1.num_edges (); e ++)
+    {
+      if (kd1.edge_smoothing_oriented % e)
+	edge_smoothing_oriented.push (e);
+    }
+  
+  for (unsigned i = 1; i <= kd2.n_crossings; i ++)
+    for (unsigned j = 1; j <= 4; j ++)
+      crossings[kd1.n_crossings + i][j] = kd1.num_epts () + kd2.crossings[i][j];
+  
+  for (unsigned e = 1; e <= kd2.num_edges (); e ++)
+    {
+      if (kd2.edge_smoothing_oriented % e)
+	edge_smoothing_oriented.push (kd1.num_edges () + e);
+    }
+  
+  // ?? break this out into aux function
+  ept_crossing = basedvector<unsigned, 1> (num_epts ());
+  ept_index = basedvector<unsigned, 1> (num_epts ());
+  for (unsigned i = 1; i <= n_crossings; i ++)
+    {
+      for (unsigned j = 1; j <= 4; j ++)
+	{
+	  unsigned p = crossings[i][j];
+	  ept_crossing[p] = i;
+	  ept_index[p] = j;
+	}
+    }
+  
+#ifndef NDEBUG
+  check_crossings ();
+#endif
+}
+
 knot_diagram::knot_diagram (mirror, const knot_diagram &kd)
   : name("mirror(" + kd.name + ")"),
     n_crossings(kd.n_crossings),
@@ -521,6 +575,63 @@ knot_diagram::rotate_crossing (unsigned c)
     }
 }
 
+unsigned
+knot_diagram::total_linking_number () const
+{
+  unionfind<1> u (num_edges ());
+  
+  for (unsigned i = 1; i <= n_crossings; i ++)
+    {
+      u.join (ept_edge (crossings[i][1]),
+	      ept_edge (crossings[i][3]));
+      u.join (ept_edge (crossings[i][2]),
+	      ept_edge (crossings[i][4]));
+    }
+  unsigned n = u.num_sets ();
+  
+  map<unsigned, unsigned> root_comp;
+  unsigned t = 0;
+  for (unsigned i = 1; i <= num_edges (); i ++)
+    {
+      if (u.find (i) == i)
+	{
+	  ++ t;
+	  root_comp.push (i, t);
+	}
+    }
+  assert (t == n);
+  
+  unsigned total_lk = 0;
+  
+  for (unsigned i = 1; i <= n; i ++)
+    for (unsigned j = i + 1; j <= n; j ++)
+      {
+	if (i == j)
+	  continue;
+	
+	int lk = 0;
+	for (unsigned x = 1; x <= n_crossings; x ++)
+	  {
+	    unsigned r1 = root_comp(u.find (ept_edge (crossings[x][1]))),
+	      r2 = root_comp(u.find (ept_edge (crossings[x][2])));
+	    if (((r1 == i) && (r2 == j))
+		|| ((r2 == i) && (r1 == j)))
+	      {	
+		if (is_to_ept (crossings[x][1]) == is_to_ept (crossings[x][4]))
+		  lk ++;
+		else
+		  lk --;
+	      }
+	  }
+	assert (is_even (lk));
+	lk /= 2;
+	
+	total_lk += abs (lk);
+      }
+  
+  return total_lk;
+}
+
 knot_diagram::knot_diagram (connect_sum,
 			    const knot_diagram &d1,
 			    const knot_diagram &d2)
diff --git a/knot_diagram.h b/knot_diagram.h
index 46247aa..e721233 100644
--- a/knot_diagram.h
+++ b/knot_diagram.h
@@ -1,4 +1,14 @@
 
+// for building knot_diagram
+inline unsigned edge_from_ept (unsigned e)
+{
+  return e * 2 - 1;
+}
+inline unsigned edge_to_ept (unsigned e)
+{
+  return e * 2;
+}
+
 static inline unsigned
 add_base1_mod4 (unsigned x, unsigned y)
 {
@@ -11,6 +21,7 @@ add_base1_mod4 (unsigned x, unsigned y)
 enum mirror { MIRROR };
 enum connect_sum { CONNECT_SUM };
 enum sublink { SUBLINK };
+enum disjoint_union { DISJOINT_UNION };
 
 class knot_diagram
 {
@@ -122,6 +133,9 @@ class knot_diagram
   knot_diagram (sublink,
 		smallbitset c,
 		const knot_diagram &kd);
+  knot_diagram (disjoint_union,
+		const knot_diagram &kd1,
+		const knot_diagram &kd2);
   
   knot_diagram (const std::string &name_, unsigned n_crossings_, unsigned crossings_ar[][4]);
   knot_diagram (const std::string &name_, const basedvector<basedvector<unsigned, 1>, 1> &crossings_);
@@ -157,6 +171,8 @@ class knot_diagram
   
   int writhe () const { return (int)nplus - (int)nminus; }
   
+  unsigned total_linking_number () const;
+  
   basedvector<basedvector<int, 1>, 1> planar_diagram_crossings () const;
   
   hash_t hash_self () const;
diff --git a/knot_parser/knot_parser.cc b/knot_parser/knot_parser.cc
index 6b764d5..9f0a338 100644
--- a/knot_parser/knot_parser.cc
+++ b/knot_parser/knot_parser.cc
@@ -1,9 +1,8 @@
-/* A Bison parser, made by GNU Bison 2.4.3.  */
+/* A Bison parser, made by GNU Bison 2.6.5.  */
 
 /* Skeleton implementation for Bison LALR(1) parsers in C++
    
-      Copyright (C) 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-   Software Foundation, Inc.
+      Copyright (C) 2002-2012 Free Software Foundation, Inc.
    
    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
@@ -34,37 +33,41 @@
 
 /* First part of user declarations.  */
 
-
-/* Line 311 of lalr1.cc  */
-#line 40 "knot_parser/knot_parser.cc"
+/* Line 278 of lalr1.cc  */
+#line 38 "knot_parser/knot_parser.cc"
 
 
 #include "knot_parser.hh"
 
 /* User implementation prologue.  */
 
-
-/* Line 317 of lalr1.cc  */
-#line 49 "knot_parser/knot_parser.cc"
+/* Line 284 of lalr1.cc  */
+#line 46 "knot_parser/knot_parser.cc"
 /* Unqualified %code blocks.  */
-
-/* Line 318 of lalr1.cc  */
+/* Line 285 of lalr1.cc  */
 #line 15 "knot_parser/knot_parser.yy"
 
 #define YY_DECL \
   yy::knot_parser::token_type knot_yylex (yy::knot_parser::semantic_type *yylval)
 YY_DECL;
 
-
-/* Line 318 of lalr1.cc  */
-#line 29 "knot_parser/knot_parser.yy"
+/* Line 285 of lalr1.cc  */
+#line 30 "knot_parser/knot_parser.yy"
 
 #define yylex knot_yylex
 
 
+/* Line 285 of lalr1.cc  */
+#line 62 "knot_parser/knot_parser.cc"
 
-/* Line 318 of lalr1.cc  */
-#line 68 "knot_parser/knot_parser.cc"
+
+# ifndef YY_NULL
+#  if defined __cplusplus && 201103L <= __cplusplus
+#   define YY_NULL nullptr
+#  else
+#   define YY_NULL 0
+#  endif
+# endif
 
 #ifndef YY_
 # if defined YYENABLE_NLS && YYENABLE_NLS
@@ -78,6 +81,27 @@ YY_DECL;
 # endif
 #endif
 
+#define YYRHSLOC(Rhs, K) ((Rhs)[K])
+/* YYLLOC_DEFAULT -- Set CURRENT to span from RHS[1] to RHS[N].
+   If N is 0, then set CURRENT to the empty location which ends
+   the previous symbol: RHS[0] (always defined).  */
+
+# ifndef YYLLOC_DEFAULT
+#  define YYLLOC_DEFAULT(Current, Rhs, N)                               \
+    do                                                                  \
+      if (N)                                                            \
+        {                                                               \
+          (Current).begin  = YYRHSLOC (Rhs, 1).begin;                   \
+          (Current).end    = YYRHSLOC (Rhs, N).end;                     \
+        }                                                               \
+      else                                                              \
+        {                                                               \
+          (Current).begin = (Current).end = YYRHSLOC (Rhs, 0).end;      \
+        }                                                               \
+    while (/*CONSTCOND*/ false)
+# endif
+
+
 /* Suppress unused-variable warnings by "using" E.  */
 #define YYUSE(e) ((void) (e))
 
@@ -128,49 +152,8 @@ do {					\
 
 
 namespace yy {
-
-/* Line 380 of lalr1.cc  */
-#line 134 "knot_parser/knot_parser.cc"
-#if YYERROR_VERBOSE
-
-  /* Return YYSTR after stripping away unnecessary quotes and
-     backslashes, so that it's suitable for yyerror.  The heuristic is
-     that double-quoting is unnecessary unless the string contains an
-     apostrophe, a comma, or backslash (other than backslash-backslash).
-     YYSTR is taken from yytname.  */
-  std::string
-  knot_parser::yytnamerr_ (const char *yystr)
-  {
-    if (*yystr == '"')
-      {
-        std::string yyr = "";
-        char const *yyp = yystr;
-
-        for (;;)
-          switch (*++yyp)
-            {
-            case '\'':
-            case ',':
-              goto do_not_strip_quotes;
-
-            case '\\':
-              if (*++yyp != '\\')
-                goto do_not_strip_quotes;
-              /* Fall through.  */
-            default:
-              yyr += *yyp;
-              break;
-
-            case '"':
-              return yyr;
-            }
-      do_not_strip_quotes: ;
-      }
-
-    return yystr;
-  }
-
-#endif
+/* Line 352 of lalr1.cc  */
+#line 157 "knot_parser/knot_parser.cc"
 
   /// Build a parser object.
   knot_parser::knot_parser (knot_diagram &parsed_knot_yyarg)
@@ -198,6 +181,9 @@ namespace yy {
   {
     YYUSE (yylocationp);
     YYUSE (yyvaluep);
+    std::ostream& yyo = debug_stream ();
+    std::ostream& yyoutput = yyo;
+    YYUSE (yyoutput);
     switch (yytype)
       {
          default:
@@ -271,6 +257,18 @@ namespace yy {
   }
 #endif
 
+  inline bool
+  knot_parser::yy_pact_value_is_default_ (int yyvalue)
+  {
+    return yyvalue == yypact_ninf_;
+  }
+
+  inline bool
+  knot_parser::yy_table_value_is_error_ (int yyvalue)
+  {
+    return yyvalue == yytable_ninf_;
+  }
+
   int
   knot_parser::parse ()
   {
@@ -278,17 +276,18 @@ namespace yy {
     int yychar = yyempty_;
     int yytoken = 0;
 
-    /* State.  */
+    // State.
     int yyn;
     int yylen = 0;
     int yystate = 0;
 
-    /* Error handling.  */
+    // Error handling.
     int yynerrs_ = 0;
     int yyerrstatus_ = 0;
 
     /// Semantic value of the lookahead.
-    semantic_type yylval;
+    static semantic_type yyval_default;
+    semantic_type yylval = yyval_default;
     /// Location of the lookahead.
     location_type yylloc;
     /// The locations where the error started and ended.
@@ -330,7 +329,7 @@ namespace yy {
 
     /* Try to take a decision without lookahead.  */
     yyn = yypact_[yystate];
-    if (yyn == yypact_ninf_)
+    if (yy_pact_value_is_default_ (yyn))
       goto yydefault;
 
     /* Read a lookahead token.  */
@@ -340,7 +339,6 @@ namespace yy {
 	yychar = yylex (&yylval);
       }
 
-
     /* Convert token to internal form.  */
     if (yychar <= yyeof_)
       {
@@ -363,8 +361,8 @@ namespace yy {
     yyn = yytable_[yyn];
     if (yyn <= 0)
       {
-	if (yyn == 0 || yyn == yytable_ninf_)
-	goto yyerrlab;
+	if (yy_table_value_is_error_ (yyn))
+	  goto yyerrlab;
 	yyn = -yyn;
 	goto yyreduce;
       }
@@ -418,17 +416,35 @@ namespace yy {
     YY_REDUCE_PRINT (yyn);
     switch (yyn)
       {
-	  case 10:
+	  case 2:
+/* Line 661 of lalr1.cc  */
+#line 60 "knot_parser/knot_parser.yy"
+    { 
+        parsed_knot = *(yysemantic_stack_[(1) - (1)].kd); 
+        delete (yysemantic_stack_[(1) - (1)].kd);
+      }
+    break;
 
-/* Line 678 of lalr1.cc  */
-#line 67 "knot_parser/knot_parser.yy"
+  case 4:
+/* Line 661 of lalr1.cc  */
+#line 69 "knot_parser/knot_parser.yy"
+    {
+       (yyval.kd) = new knot_diagram (DISJOINT_UNION, *(yysemantic_stack_[(2) - (1)].kd), *(yysemantic_stack_[(2) - (2)].kd));
+       delete (yysemantic_stack_[(2) - (1)].kd);
+       delete (yysemantic_stack_[(2) - (2)].kd);
+     }
+    break;
+
+  case 13:
+/* Line 661 of lalr1.cc  */
+#line 89 "knot_parser/knot_parser.yy"
     { 
 	unsigned n = (yysemantic_stack_[(3) - (1)].integer),
 	  k = (yysemantic_stack_[(3) - (3)].integer);
 	
 	if (n >= 1 && n <= 10
 	    && k >= 1 && k <= rolfsen_crossing_knots (n))
-	  parsed_knot = knot_diagram (rolfsen_knot (n, k));
+	  (yyval.kd) = new knot_diagram (rolfsen_knot (n, k));
 	else
 	  {
 	    fprintf (stderr, "knot_parser: no such Rolfsen knot `%d_%d'\n",
@@ -438,10 +454,9 @@ namespace yy {
       }
     break;
 
-  case 11:
-
-/* Line 678 of lalr1.cc  */
-#line 85 "knot_parser/knot_parser.yy"
+  case 14:
+/* Line 661 of lalr1.cc  */
+#line 107 "knot_parser/knot_parser.yy"
     {
 	unsigned n = (yysemantic_stack_[(3) - (1)].integer),
 	  k = (yysemantic_stack_[(3) - (3)].integer);
@@ -449,7 +464,7 @@ namespace yy {
 	
 	if (n >= 1 && n <= 16
 	    && k >= 1 && k <= htw_knots (n, alt))
-	  parsed_knot = knot_diagram (htw_knot (n, alt, k));
+	  (yyval.kd) = new knot_diagram (htw_knot (n, alt, k));
 	else
 	  {
 	    fprintf (stderr, "knot_parser: no such HTW knot `%d%c%d'\n",
@@ -459,10 +474,9 @@ namespace yy {
       }
     break;
 
-  case 12:
-
-/* Line 678 of lalr1.cc  */
-#line 104 "knot_parser/knot_parser.yy"
+  case 15:
+/* Line 661 of lalr1.cc  */
+#line 126 "knot_parser/knot_parser.yy"
     {
 	unsigned n = (yysemantic_stack_[(4) - (2)].integer),
 	  k = (yysemantic_stack_[(4) - (4)].integer);
@@ -470,7 +484,7 @@ namespace yy {
 	
 	if (n >= 1 && n <= 14
 	    && k >= 1 && k <= mt_links (n, alt))
-	  parsed_knot = knot_diagram (mt_link (n, alt, k));
+	  (yyval.kd) = new knot_diagram (mt_link (n, alt, k));
 	else
 	  {
 	    fprintf (stderr, "knot_parser: no such MT link `%d%c%d'\n", 
@@ -480,75 +494,66 @@ namespace yy {
       }
     break;
 
-  case 13:
-
-/* Line 678 of lalr1.cc  */
-#line 123 "knot_parser/knot_parser.yy"
-    { parsed_knot = knot_diagram (planar_diagram ("<parsed>", *(yysemantic_stack_[(4) - (3)].int_vec2))); }
-    break;
-
-  case 14:
-
-/* Line 678 of lalr1.cc  */
-#line 125 "knot_parser/knot_parser.yy"
-    { parsed_knot = knot_diagram (planar_diagram ("<parsed>", *(yysemantic_stack_[(4) - (3)].int_vec2))); }
-    break;
-
-  case 15:
-
-/* Line 678 of lalr1.cc  */
-#line 130 "knot_parser/knot_parser.yy"
-    {
-	basedvector<basedvector<int, 1>, 1> even_labels (1);
-	even_labels[1] = *(yysemantic_stack_[(4) - (3)].int_vec);
-	parsed_knot = knot_diagram (dt_code ("<parsed>", even_labels));
-      }
-    break;
-
   case 16:
-
-/* Line 678 of lalr1.cc  */
-#line 136 "knot_parser/knot_parser.yy"
-    { parsed_knot = knot_diagram (dt_code ("<parsed>", *(yysemantic_stack_[(4) - (3)].int_vec2))); }
+/* Line 661 of lalr1.cc  */
+#line 145 "knot_parser/knot_parser.yy"
+    { (yyval.kd) = new knot_diagram (planar_diagram ("<parsed>", *(yysemantic_stack_[(4) - (3)].int_vec2))); }
     break;
 
   case 17:
-
-/* Line 678 of lalr1.cc  */
-#line 138 "knot_parser/knot_parser.yy"
-    { parsed_knot = knot_diagram (dt_code ("<parsed>", (yysemantic_stack_[(4) - (3)].string))); }
+/* Line 661 of lalr1.cc  */
+#line 147 "knot_parser/knot_parser.yy"
+    { (yyval.kd) = new knot_diagram (planar_diagram ("<parsed>", *(yysemantic_stack_[(4) - (3)].int_vec2))); }
     break;
 
   case 18:
-
-/* Line 678 of lalr1.cc  */
-#line 143 "knot_parser/knot_parser.yy"
-    { parsed_knot = knot_diagram (torus_knot ((yysemantic_stack_[(6) - (3)].integer), (yysemantic_stack_[(6) - (5)].integer))); }
+/* Line 661 of lalr1.cc  */
+#line 152 "knot_parser/knot_parser.yy"
+    {
+	basedvector<basedvector<int, 1>, 1> even_labels (1);
+	even_labels[1] = *(yysemantic_stack_[(4) - (3)].int_vec);
+	(yyval.kd) = new knot_diagram (dt_code ("<parsed>", even_labels));
+      }
     break;
 
   case 19:
-
-/* Line 678 of lalr1.cc  */
-#line 148 "knot_parser/knot_parser.yy"
-    { parsed_knot = knot_diagram (braid ((yysemantic_stack_[(6) - (3)].integer), *(yysemantic_stack_[(6) - (5)].int_vec))); }
+/* Line 661 of lalr1.cc  */
+#line 158 "knot_parser/knot_parser.yy"
+    { (yyval.kd) = new knot_diagram (dt_code ("<parsed>", *(yysemantic_stack_[(4) - (3)].int_vec2))); }
     break;
 
   case 20:
+/* Line 661 of lalr1.cc  */
+#line 160 "knot_parser/knot_parser.yy"
+    { (yyval.kd) = new knot_diagram (dt_code ("<parsed>", (yysemantic_stack_[(4) - (3)].string))); }
+    break;
 
-/* Line 678 of lalr1.cc  */
-#line 153 "knot_parser/knot_parser.yy"
+  case 21:
+/* Line 661 of lalr1.cc  */
+#line 165 "knot_parser/knot_parser.yy"
+    { (yyval.kd) = new knot_diagram (torus_knot ((yysemantic_stack_[(6) - (3)].integer), (yysemantic_stack_[(6) - (5)].integer))); }
+    break;
+
+  case 22:
+/* Line 661 of lalr1.cc  */
+#line 170 "knot_parser/knot_parser.yy"
+    { (yyval.kd) = new knot_diagram (braid ((yysemantic_stack_[(6) - (3)].integer), *(yysemantic_stack_[(6) - (5)].int_vec))); }
+    break;
+
+  case 23:
+/* Line 661 of lalr1.cc  */
+#line 175 "knot_parser/knot_parser.yy"
     {
 	unsigned unknot_ar[1][4] = {
 	  { 2, 1, 3, 4, },
 	};
-	parsed_knot = knot_diagram ("U", 1, unknot_ar);
+	(yyval.kd) = new knot_diagram ("U", 1, unknot_ar);
       }
     break;
 
-  case 23:
-
-/* Line 678 of lalr1.cc  */
-#line 168 "knot_parser/knot_parser.yy"
+  case 26:
+/* Line 661 of lalr1.cc  */
+#line 190 "knot_parser/knot_parser.yy"
     {
 	basedvector<basedvector<int, 1>, 1> *v
 	  = new basedvector<basedvector<int, 1>, 1> ();
@@ -557,10 +562,9 @@ namespace yy {
       }
     break;
 
-  case 24:
-
-/* Line 678 of lalr1.cc  */
-#line 175 "knot_parser/knot_parser.yy"
+  case 27:
+/* Line 661 of lalr1.cc  */
+#line 197 "knot_parser/knot_parser.yy"
     { 
 	basedvector<basedvector<int, 1>, 1> *v = (yysemantic_stack_[(3) - (1)].int_vec2);
 	v->append (*(yysemantic_stack_[(3) - (3)].int_vec));
@@ -568,24 +572,21 @@ namespace yy {
       }
     break;
 
-  case 25:
-
-/* Line 678 of lalr1.cc  */
-#line 184 "knot_parser/knot_parser.yy"
+  case 28:
+/* Line 661 of lalr1.cc  */
+#line 206 "knot_parser/knot_parser.yy"
     { (yyval.int_vec) = (yysemantic_stack_[(3) - (2)].int_vec); }
     break;
 
-  case 26:
-
-/* Line 678 of lalr1.cc  */
-#line 186 "knot_parser/knot_parser.yy"
+  case 29:
+/* Line 661 of lalr1.cc  */
+#line 208 "knot_parser/knot_parser.yy"
     { (yyval.int_vec) = (yysemantic_stack_[(3) - (2)].int_vec); }
     break;
 
-  case 27:
-
-/* Line 678 of lalr1.cc  */
-#line 191 "knot_parser/knot_parser.yy"
+  case 30:
+/* Line 661 of lalr1.cc  */
+#line 213 "knot_parser/knot_parser.yy"
     {
 	basedvector<int, 1> *v =
 	  new basedvector<int, 1> ();
@@ -594,10 +595,9 @@ namespace yy {
       }
     break;
 
-  case 28:
-
-/* Line 678 of lalr1.cc  */
-#line 198 "knot_parser/knot_parser.yy"
+  case 31:
+/* Line 661 of lalr1.cc  */
+#line 220 "knot_parser/knot_parser.yy"
     {
 	basedvector<int, 1> *v = (yysemantic_stack_[(3) - (1)].int_vec);
 	v->append ((yysemantic_stack_[(3) - (3)].integer));
@@ -605,10 +605,9 @@ namespace yy {
       }
     break;
 
-  case 29:
-
-/* Line 678 of lalr1.cc  */
-#line 207 "knot_parser/knot_parser.yy"
+  case 32:
+/* Line 661 of lalr1.cc  */
+#line 229 "knot_parser/knot_parser.yy"
     {
 	basedvector<basedvector<int, 1>, 1> *v
 	  = new basedvector<basedvector<int, 1>, 1> ();
@@ -617,10 +616,9 @@ namespace yy {
       }
     break;
 
-  case 30:
-
-/* Line 678 of lalr1.cc  */
-#line 214 "knot_parser/knot_parser.yy"
+  case 33:
+/* Line 661 of lalr1.cc  */
+#line 236 "knot_parser/knot_parser.yy"
     { 
 	basedvector<basedvector<int, 1>, 1> *v = (yysemantic_stack_[(3) - (1)].int_vec2);
 	v->append (*(yysemantic_stack_[(3) - (3)].int_vec));
@@ -628,10 +626,9 @@ namespace yy {
       }
     break;
 
-  case 31:
-
-/* Line 678 of lalr1.cc  */
-#line 223 "knot_parser/knot_parser.yy"
+  case 34:
+/* Line 661 of lalr1.cc  */
+#line 245 "knot_parser/knot_parser.yy"
     {
 	basedvector<int, 1> *v
 	  = new basedvector<int, 1> ();
@@ -644,12 +641,22 @@ namespace yy {
     break;
 
 
-
-/* Line 678 of lalr1.cc  */
-#line 650 "knot_parser/knot_parser.cc"
+/* Line 661 of lalr1.cc  */
+#line 646 "knot_parser/knot_parser.cc"
 	default:
           break;
       }
+    /* User semantic actions sometimes alter yychar, and that requires
+       that yytoken be updated with the new translation.  We take the
+       approach of translating immediately before every use of yytoken.
+       One alternative is translating here after every semantic action,
+       but that translation would be missed if the semantic action
+       invokes YYABORT, YYACCEPT, or YYERROR immediately after altering
+       yychar.  In the case of YYABORT or YYACCEPT, an incorrect
+       destructor might then be invoked immediately.  In the case of
+       YYERROR, subsequent parser actions might lead to an incorrect
+       destructor call or verbose syntax error message before the
+       lookahead is translated.  */
     YY_SYMBOL_PRINT ("-> $$ =", yyr1_[yyn], &yyval, &yyloc);
 
     yypop_ (yylen);
@@ -673,11 +680,17 @@ namespace yy {
   | yyerrlab -- here on detecting error |
   `------------------------------------*/
   yyerrlab:
+    /* Make sure we have latest lookahead translation.  See comments at
+       user semantic actions for why this is necessary.  */
+    yytoken = yytranslate_ (yychar);
+
     /* If not already recovering from an error, report this error.  */
     if (!yyerrstatus_)
       {
 	++yynerrs_;
-	error (yylloc, yysyntax_error_ (yystate));
+	if (yychar == yyempty_)
+	  yytoken = yyempty_;
+	error (yylloc, yysyntax_error_ (yystate, yytoken));
       }
 
     yyerror_range[1] = yylloc;
@@ -732,7 +745,7 @@ namespace yy {
     for (;;)
       {
 	yyn = yypact_[yystate];
-	if (yyn != yypact_ninf_)
+	if (!yy_pact_value_is_default_ (yyn))
 	{
 	  yyn += yyterror_;
 	  if (0 <= yyn && yyn <= yylast_ && yycheck_[yyn] == yyterror_)
@@ -782,7 +795,13 @@ namespace yy {
 
   yyreturn:
     if (yychar != yyempty_)
-      yydestruct_ ("Cleanup: discarding lookahead", yytoken, &yylval, &yylloc);
+      {
+        /* Make sure we have latest lookahead translation.  See comments
+           at user semantic actions for why this is necessary.  */
+        yytoken = yytranslate_ (yychar);
+        yydestruct_ ("Cleanup: discarding lookahead", yytoken, &yylval,
+                     &yylloc);
+      }
 
     /* Do not reclaim the symbols of the rule which action triggered
        this YYABORT or YYACCEPT.  */
@@ -801,117 +820,76 @@ namespace yy {
 
   // Generate an error message.
   std::string
-  knot_parser::yysyntax_error_ (int yystate)
+  knot_parser::yysyntax_error_ (int, int)
   {
-    std::string res;
-    YYUSE (yystate);
-#if YYERROR_VERBOSE
-    int yyn = yypact_[yystate];
-    if (yypact_ninf_ < yyn && yyn <= yylast_)
-      {
-	/* Start YYX at -YYN if negative to avoid negative indexes in
-	   YYCHECK.  */
-	int yyxbegin = yyn < 0 ? -yyn : 0;
-
-	/* Stay within bounds of both yycheck and yytname.  */
-	int yychecklim = yylast_ - yyn + 1;
-	int yyxend = yychecklim < yyntokens_ ? yychecklim : yyntokens_;
-	int count = 0;
-	for (int x = yyxbegin; x < yyxend; ++x)
-	  if (yycheck_[x + yyn] == x && x != yyterror_)
-	    ++count;
-
-	// FIXME: This method of building the message is not compatible
-	// with internationalization.  It should work like yacc.c does it.
-	// That is, first build a string that looks like this:
-	// "syntax error, unexpected %s or %s or %s"
-	// Then, invoke YY_ on this string.
-	// Finally, use the string as a format to output
-	// yytname_[tok], etc.
-	// Until this gets fixed, this message appears in English only.
-	res = "syntax error, unexpected ";
-	res += yytnamerr_ (yytname_[tok]);
-	if (count < 5)
-	  {
-	    count = 0;
-	    for (int x = yyxbegin; x < yyxend; ++x)
-	      if (yycheck_[x + yyn] == x && x != yyterror_)
-		{
-		  res += (!count++) ? ", expecting " : " or ";
-		  res += yytnamerr_ (yytname_[x]);
-		}
-	  }
-      }
-    else
-#endif
-      res = YY_("syntax error");
-    return res;
+    return YY_("syntax error");
   }
 
 
   /* YYPACT[STATE-NUM] -- Index in YYTABLE of the portion describing
      STATE-NUM.  */
-  const signed char knot_parser::yypact_ninf_ = -36;
+  const signed char knot_parser::yypact_ninf_ = -39;
   const signed char
   knot_parser::yypact_[] =
   {
-        -1,    10,    -9,    25,    26,    37,     9,   -36,    43,   -36,
-     -36,   -36,   -36,   -36,   -36,   -36,   -36,   -36,   -36,    41,
-      42,    -7,    -3,    44,    24,    45,   -36,   -36,   -36,    30,
-      46,    46,     7,   -36,    11,   -36,   -36,    33,    12,    15,
-      -4,    48,    34,    49,    16,    -2,   -36,     0,   -36,    47,
-     -36,   -36,   -36,    51,     0,   -36,    53,    38,   -36,   -36,
-     -36,   -36,   -36,    50,    39,    55,   -36,   -36,    52,    57,
-      54,    58,    59,   -36
+        -1,    10,    -9,    25,    26,    37,     9,   -39,    43,    -1,
+     -39,   -39,   -39,   -39,   -39,   -39,   -39,   -39,   -39,   -39,
+     -39,    41,    42,    -7,    -3,    44,    24,    45,   -39,    -1,
+     -39,   -39,    30,    46,    46,     7,   -39,    11,   -39,   -39,
+      33,    12,    15,    -4,    48,    34,    49,    16,    -2,   -39,
+       0,   -39,    47,   -39,   -39,   -39,    51,     0,   -39,    53,
+      38,   -39,   -39,   -39,   -39,   -39,    50,    39,    55,   -39,
+     -39,    52,    57,    54,    58,    59,   -39
   };
 
-  /* YYDEFACT[S] -- default rule to reduce with in state S when YYTABLE
-     doesn't specify something else to do.  Zero means the default is an
-     error.  */
+  /* YYDEFACT[S] -- default reduction number in state S.  Performed when
+     YYTABLE doesn't specify something else to do.  Zero means the
+     default is an error.  */
   const unsigned char
   knot_parser::yydefact_[] =
   {
-         0,     0,     0,     0,     0,     0,     0,    20,     0,     2,
-       3,     4,     5,     6,     7,     9,     8,    21,    22,     0,
-       0,     0,     0,     0,     0,     0,     1,    10,    11,     0,
-       0,     0,     0,    23,     0,    29,    27,     0,     0,     0,
-       0,     0,     0,     0,     0,     0,    13,     0,    14,     0,
-      17,    16,    15,     0,     0,    12,     0,     0,    26,    25,
-      24,    30,    28,     0,     0,     0,    19,    18,     0,     0,
-       0,     0,     0,    31
+         0,     0,     0,     0,     0,     0,     0,    23,     0,     2,
+       3,     5,     6,     7,     8,     9,    10,    12,    11,    24,
+      25,     0,     0,     0,     0,     0,     0,     0,     1,     4,
+      13,    14,     0,     0,     0,     0,    26,     0,    32,    30,
+       0,     0,     0,     0,     0,     0,     0,     0,     0,    16,
+       0,    17,     0,    20,    19,    18,     0,     0,    15,     0,
+       0,    29,    28,    27,    33,    31,     0,     0,     0,    22,
+      21,     0,     0,     0,     0,     0,    34
   };
 
   /* YYPGOTO[NTERM-NUM].  */
   const signed char
   knot_parser::yypgoto_[] =
   {
-       -36,   -36,   -36,   -36,   -36,   -36,   -36,   -36,   -36,   -36,
-      40,    56,   -35,     8,   -36,    13
+       -39,   -39,    62,   -39,   -39,   -39,   -39,   -39,   -39,   -39,
+     -39,   -39,    40,    56,   -38,     5,   -39,    13
   };
 
   /* YYDEFGOTO[NTERM-NUM].  */
   const signed char
   knot_parser::yydefgoto_[] =
   {
-        -1,     8,     9,    10,    11,    12,    13,    14,    15,    16,
-      20,    32,    33,    39,    34,    35
+        -1,     8,    29,    10,    11,    12,    13,    14,    15,    16,
+      17,    18,    22,    35,    36,    42,    37,    38
   };
 
   /* YYTABLE[YYPACT[STATE-NUM]].  What to do in state STATE-NUM.  If
      positive, shift that token.  If negative, reduce the rule which
-     number is the opposite.  If zero, do what YYDEFACT says.  */
+     number is the opposite.  If YYTABLE_NINF_, syntax error.  */
   const signed char knot_parser::yytable_ninf_ = -1;
   const unsigned char
   knot_parser::yytable_[] =
   {
-        36,    29,     1,     2,     3,     4,     5,    21,     6,    30,
-       7,    37,    60,    30,    31,    54,    30,    53,    31,    63,
-      59,    31,    17,    18,    46,    19,    47,    25,    48,    51,
-      49,    47,    52,    58,    53,    53,    17,    18,    44,    45,
-      24,    22,    23,    26,    27,    28,    43,    40,    42,    36,
-      50,    55,    57,    56,    62,    29,    64,    65,    68,    67,
-      70,    72,    61,     0,    41,     0,     0,    66,     0,     0,
-       0,    69,     0,    71,     0,     0,    73,     0,    38
+        39,    32,     1,     2,     3,     4,     5,    23,     6,    33,
+       7,    40,    63,    33,    34,    57,    33,    56,    34,    66,
+      62,    34,    19,    20,    49,    21,    50,    27,    51,    54,
+      52,    50,    55,    61,    56,    56,    19,    20,    47,    48,
+      26,    24,    25,    28,    30,    31,    46,    43,    45,    39,
+      53,    58,    60,    59,    65,    32,    67,    68,    71,    70,
+      73,    75,     9,     0,     0,    64,    44,    69,     0,     0,
+       0,    72,     0,    74,     0,     0,    76,     0,     0,     0,
+      41
   };
 
   /* YYCHECK.  */
@@ -919,13 +897,14 @@ namespace yy {
   knot_parser::yycheck_[] =
   {
          3,     8,     3,     4,     5,     6,     7,    16,     9,    16,
-      11,    14,    47,    16,    21,    19,    16,    19,    21,    54,
+      11,    14,    50,    16,    21,    19,    16,    19,    21,    57,
       22,    21,    12,    13,    17,    15,    19,    18,    17,    17,
-      19,    19,    17,    17,    19,    19,    12,    13,    30,    31,
+      19,    19,    17,    17,    19,    19,    12,    13,    33,    34,
        3,    16,    16,     0,     3,     3,    16,     3,     3,     3,
       17,     3,     3,    19,     3,     8,     3,    19,     3,    20,
-       3,     3,    49,    -1,    24,    -1,    -1,    17,    -1,    -1,
-      -1,    19,    -1,    19,    -1,    -1,    17,    -1,    22
+       3,     3,     0,    -1,    -1,    52,    26,    17,    -1,    -1,
+      -1,    19,    -1,    19,    -1,    -1,    17,    -1,    -1,    -1,
+      24
   };
 
   /* STOS_[STATE-NUM] -- The (internal number of the) accessing
@@ -934,13 +913,13 @@ namespace yy {
   knot_parser::yystos_[] =
   {
          0,     3,     4,     5,     6,     7,     9,    11,    24,    25,
-      26,    27,    28,    29,    30,    31,    32,    12,    13,    15,
-      33,    16,    16,    16,     3,    18,     0,     3,     3,     8,
-      16,    21,    34,    35,    37,    38,     3,    14,    34,    36,
-       3,    33,     3,    16,    36,    36,    17,    19,    17,    19,
-      17,    17,    17,    19,    19,     3,    19,     3,    17,    22,
-      35,    38,     3,    35,     3,    19,    17,    20,     3,    19,
-       3,    19,     3,    17
+      26,    27,    28,    29,    30,    31,    32,    33,    34,    12,
+      13,    15,    35,    16,    16,    16,     3,    18,     0,    25,
+       3,     3,     8,    16,    21,    36,    37,    39,    40,     3,
+      14,    36,    38,     3,    35,     3,    16,    38,    38,    17,
+      19,    17,    19,    17,    17,    17,    19,    19,     3,    19,
+       3,    17,    22,    37,    40,     3,    37,     3,    19,    17,
+      20,     3,    19,     3,    19,     3,    17
   };
 
 #if YYDEBUG
@@ -959,23 +938,23 @@ namespace yy {
   const unsigned char
   knot_parser::yyr1_[] =
   {
-         0,    23,    24,    24,    24,    24,    24,    24,    24,    24,
-      25,    26,    27,    28,    28,    29,    29,    29,    30,    31,
-      32,    33,    33,    34,    34,    35,    35,    36,    36,    37,
-      37,    38
+         0,    23,    24,    25,    25,    26,    26,    26,    26,    26,
+      26,    26,    26,    27,    28,    29,    30,    30,    31,    31,
+      31,    32,    33,    34,    35,    35,    36,    36,    37,    37,
+      38,    38,    39,    39,    40
   };
 
   /* YYR2[YYN] -- Number of symbols composing right hand side of rule YYN.  */
   const unsigned char
   knot_parser::yyr2_[] =
   {
-         0,     2,     1,     1,     1,     1,     1,     1,     1,     1,
-       3,     3,     4,     4,     4,     4,     4,     4,     6,     6,
-       1,     1,     1,     1,     3,     3,     3,     1,     3,     1,
-       3,    10
+         0,     2,     1,     1,     2,     1,     1,     1,     1,     1,
+       1,     1,     1,     3,     3,     4,     4,     4,     4,     4,
+       4,     6,     6,     1,     1,     1,     1,     3,     3,     3,
+       1,     3,     1,     3,    10
   };
 
-#if YYDEBUG || YYERROR_VERBOSE || YYTOKEN_TABLE
+#if YYDEBUG
   /* YYTNAME[SYMBOL-NUM] -- String name of the symbol SYMBOL-NUM.
      First, the terminals, then, starting at \a yyntokens_, nonterminals.  */
   const char*
@@ -983,30 +962,30 @@ namespace yy {
   {
     "\"end of file\"", "error", "$undefined", "INT", "PD", "DT", "BR",
   "LINK", "X", "T", "U", "UNKNOT", "ALT", "NONALT", "STRING", "'_'", "'['",
-  "']'", "'('", "','", "')'", "'{'", "'}'", "$accept", "knot",
-  "rolfsen_knot", "htw_knot", "mt_link", "planar_diagram", "dt",
+  "']'", "'('", "','", "')'", "'{'", "'}'", "$accept", "entry", "knot",
+  "knot_1", "rolfsen_knot", "htw_knot", "mt_link", "planar_diagram", "dt",
   "torus_link", "braid", "unknot", "alt_spec", "int_vec2", "int_vec",
-  "int_vec_1", "crossing_vec", "crossing", 0
+  "int_vec_1", "crossing_vec", "crossing", YY_NULL
   };
-#endif
 
-#if YYDEBUG
+
   /* YYRHS -- A `-1'-separated list of the rules' RHS.  */
   const knot_parser::rhs_number_type
   knot_parser::yyrhs_[] =
   {
-        24,     0,    -1,    25,    -1,    26,    -1,    27,    -1,    28,
-      -1,    29,    -1,    30,    -1,    32,    -1,    31,    -1,     3,
-      15,     3,    -1,     3,    33,     3,    -1,     7,     3,    33,
-       3,    -1,     4,    16,    34,    17,    -1,     4,    16,    37,
-      17,    -1,     5,    16,    36,    17,    -1,     5,    16,    34,
-      17,    -1,     5,    16,    14,    17,    -1,     9,    18,     3,
-      19,     3,    20,    -1,     6,    16,     3,    19,    35,    17,
-      -1,    11,    -1,    12,    -1,    13,    -1,    35,    -1,    34,
-      19,    35,    -1,    21,    36,    22,    -1,    16,    36,    17,
-      -1,     3,    -1,    36,    19,     3,    -1,    38,    -1,    37,
-      19,    38,    -1,     8,    16,     3,    19,     3,    19,     3,
-      19,     3,    17,    -1
+        24,     0,    -1,    25,    -1,    26,    -1,    25,    25,    -1,
+      27,    -1,    28,    -1,    29,    -1,    30,    -1,    31,    -1,
+      32,    -1,    34,    -1,    33,    -1,     3,    15,     3,    -1,
+       3,    35,     3,    -1,     7,     3,    35,     3,    -1,     4,
+      16,    36,    17,    -1,     4,    16,    39,    17,    -1,     5,
+      16,    38,    17,    -1,     5,    16,    36,    17,    -1,     5,
+      16,    14,    17,    -1,     9,    18,     3,    19,     3,    20,
+      -1,     6,    16,     3,    19,    37,    17,    -1,    11,    -1,
+      12,    -1,    13,    -1,    37,    -1,    36,    19,    37,    -1,
+      21,    38,    22,    -1,    16,    38,    17,    -1,     3,    -1,
+      38,    19,     3,    -1,    40,    -1,    39,    19,    40,    -1,
+       8,    16,     3,    19,     3,    19,     3,    19,     3,    17,
+      -1
   };
 
   /* YYPRHS[YYN] -- Index of the first RHS symbol of rule number YYN in
@@ -1014,20 +993,20 @@ namespace yy {
   const unsigned char
   knot_parser::yyprhs_[] =
   {
-         0,     0,     3,     5,     7,     9,    11,    13,    15,    17,
-      19,    23,    27,    32,    37,    42,    47,    52,    57,    64,
-      71,    73,    75,    77,    79,    83,    87,    91,    93,    97,
-      99,   103
+         0,     0,     3,     5,     7,    10,    12,    14,    16,    18,
+      20,    22,    24,    26,    30,    34,    39,    44,    49,    54,
+      59,    64,    71,    78,    80,    82,    84,    86,    90,    94,
+      98,   100,   104,   106,   110
   };
 
   /* YYRLINE[YYN] -- Source line where rule number YYN was defined.  */
   const unsigned char
   knot_parser::yyrline_[] =
   {
-         0,    55,    55,    56,    57,    58,    59,    60,    61,    62,
-      66,    84,   103,   122,   124,   129,   135,   137,   142,   147,
-     152,   162,   163,   167,   174,   183,   185,   190,   197,   206,
-     213,   222
+         0,    59,    59,    67,    68,    77,    78,    79,    80,    81,
+      82,    83,    84,    88,   106,   125,   144,   146,   151,   157,
+     159,   164,   169,   174,   184,   185,   189,   196,   205,   207,
+     212,   219,   228,   235,   244
   };
 
   // Print the state stack on the debug stream.
@@ -1102,10 +1081,10 @@ namespace yy {
   }
 
   const int knot_parser::yyeof_ = 0;
-  const int knot_parser::yylast_ = 78;
-  const int knot_parser::yynnts_ = 16;
+  const int knot_parser::yylast_ = 80;
+  const int knot_parser::yynnts_ = 18;
   const int knot_parser::yyempty_ = -2;
-  const int knot_parser::yyfinal_ = 26;
+  const int knot_parser::yyfinal_ = 28;
   const int knot_parser::yyterror_ = 1;
   const int knot_parser::yyerrcode_ = 256;
   const int knot_parser::yyntokens_ = 23;
@@ -1115,13 +1094,10 @@ namespace yy {
 
 
 } // yy
-
-/* Line 1054 of lalr1.cc  */
-#line 1121 "knot_parser/knot_parser.cc"
-
-
-/* Line 1056 of lalr1.cc  */
-#line 233 "knot_parser/knot_parser.yy"
+/* Line 1106 of lalr1.cc  */
+#line 1099 "knot_parser/knot_parser.cc"
+/* Line 1107 of lalr1.cc  */
+#line 255 "knot_parser/knot_parser.yy"
 
 
 void
@@ -1145,4 +1121,3 @@ parse_knot (const char *s)
   
   return d;
 }
-
diff --git a/knot_parser/knot_parser.hh b/knot_parser/knot_parser.hh
index b604953..7f07eb3 100644
--- a/knot_parser/knot_parser.hh
+++ b/knot_parser/knot_parser.hh
@@ -1,9 +1,8 @@
-/* A Bison parser, made by GNU Bison 2.4.3.  */
+/* A Bison parser, made by GNU Bison 2.6.5.  */
 
 /* Skeleton interface for Bison LALR(1) parsers in C++
    
-      Copyright (C) 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-   Software Foundation, Inc.
+      Copyright (C) 2002-2012 Free Software Foundation, Inc.
    
    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
@@ -31,41 +30,30 @@
    This special exception was added by the Free Software Foundation in
    version 2.2 of Bison.  */
 
+/**
+ ** \file knot_parser/knot_parser.hh
+ ** Define the yy::parser class.
+ */
+
 /* C++ LALR(1) parser skeleton written by Akim Demaille.  */
 
-#ifndef PARSER_HEADER_H
-# define PARSER_HEADER_H
+#ifndef YY_YY_KNOT_PARSER_KNOT_PARSER_HH_INCLUDED
+# define YY_YY_KNOT_PARSER_KNOT_PARSER_HH_INCLUDED
 
 /* "%code requires" blocks.  */
-
-/* Line 35 of lalr1.cc  */
+/* Line 36 of lalr1.cc  */
 #line 11 "knot_parser/knot_parser.yy"
 
 #include <knotkit.h>
 
 
-
-/* Line 35 of lalr1.cc  */
-#line 50 "knot_parser/knot_parser.hh"
+/* Line 36 of lalr1.cc  */
+#line 52 "knot_parser/knot_parser.hh"
 
 
 #include <string>
 #include <iostream>
 #include "stack.hh"
-
-
-namespace yy {
-
-/* Line 35 of lalr1.cc  */
-#line 61 "knot_parser/knot_parser.hh"
-  class position;
-  class location;
-
-} // yy
-
-/* Line 35 of lalr1.cc  */
-#line 68 "knot_parser/knot_parser.hh"
-
 #include "location.hh"
 
 /* Enabling traces.  */
@@ -73,43 +61,10 @@ namespace yy {
 # define YYDEBUG 0
 #endif
 
-/* Enabling verbose error messages.  */
-#ifdef YYERROR_VERBOSE
-# undef YYERROR_VERBOSE
-# define YYERROR_VERBOSE 1
-#else
-# define YYERROR_VERBOSE 0
-#endif
-
-/* Enabling the token table.  */
-#ifndef YYTOKEN_TABLE
-# define YYTOKEN_TABLE 0
-#endif
-
-/* YYLLOC_DEFAULT -- Set CURRENT to span from RHS[1] to RHS[N].
-   If N is 0, then set CURRENT to the empty location which ends
-   the previous symbol: RHS[0] (always defined).  */
-
-#ifndef YYLLOC_DEFAULT
-# define YYLLOC_DEFAULT(Current, Rhs, N)		\
-do {							\
-  if (N)						\
-    {							\
-      (Current).begin = (Rhs)[1].begin;			\
-      (Current).end   = (Rhs)[N].end;			\
-    }							\
-  else							\
-    {							\
-      (Current).begin = (Current).end = (Rhs)[0].end;	\
-    }							\
-} while (false)
-#endif
-
 
 namespace yy {
-
-/* Line 35 of lalr1.cc  */
-#line 113 "knot_parser/knot_parser.hh"
+/* Line 36 of lalr1.cc  */
+#line 68 "knot_parser/knot_parser.hh"
 
   /// A Bison parser.
   class knot_parser
@@ -119,8 +74,7 @@ namespace yy {
 #ifndef YYSTYPE
     union semantic_type
     {
-
-/* Line 35 of lalr1.cc  */
+/* Line 36 of lalr1.cc  */
 #line 21 "knot_parser/knot_parser.yy"
 
   int integer;
@@ -128,11 +82,11 @@ namespace yy {
   basedvector<int, 1> *int_vec;
   basedvector<basedvector<int, 1>, 1> *int_vec2;
   const char *string;
+  knot_diagram *kd;
 
 
-
-/* Line 35 of lalr1.cc  */
-#line 136 "knot_parser/knot_parser.hh"
+/* Line 36 of lalr1.cc  */
+#line 90 "knot_parser/knot_parser.hh"
     };
 #else
     typedef YYSTYPE semantic_type;
@@ -194,7 +148,7 @@ namespace yy {
     /// Generate an error message.
     /// \param state   the state where the error occurred.
     /// \param tok     the lookahead token.
-    virtual std::string yysyntax_error_ (int yystate);
+    virtual std::string yysyntax_error_ (int yystate, int tok);
 
 #if YYDEBUG
     /// \brief Report a symbol value on the debug stream.
@@ -230,6 +184,14 @@ namespace yy {
     /// The location stack.
     location_stack_type yylocation_stack_;
 
+    /// Whether the given \c yypact_ value indicates a defaulted state.
+    /// \param yyvalue   the value to check
+    static bool yy_pact_value_is_default_ (int yyvalue);
+
+    /// Whether the given \c yytable_ value indicates a syntax error.
+    /// \param yyvalue   the value to check
+    static bool yy_table_value_is_error_ (int yyvalue);
+
     /// Internal symbol numbers.
     typedef unsigned char token_number_type;
     /* Tables.  */
@@ -237,7 +199,7 @@ namespace yy {
     static const signed char yypact_[];
     static const signed char yypact_ninf_;
 
-    /// For a state, default rule to reduce.
+    /// For a state, default reduction number.
     /// Unless\a  yytable_ specifies something else to do.
     /// Zero means the default is an error.
     static const unsigned char yydefact_[];
@@ -261,19 +223,12 @@ namespace yy {
     /// For a rule, its LHS.
     static const unsigned char yyr1_[];
     /// For a rule, its RHS length.
-    static const unsigned char yyr2_[];
-
-#if YYDEBUG || YYERROR_VERBOSE || YYTOKEN_TABLE
-    /// For a symbol, its name in clear.
-    static const char* const yytname_[];
-#endif
-
-#if YYERROR_VERBOSE
-    /// Convert the symbol name \a n to a form suitable for a diagnostic.
-    virtual std::string yytnamerr_ (const char *n);
-#endif
+    static const unsigned char yyr2_[]; 
 
 #if YYDEBUG
+    /// For a symbol, its name in clear.
+    static const char* const yytname_[];
+
     /// A type to store symbol numbers and -1.
     typedef signed char rhs_number_type;
     /// A `-1'-separated list of the rules' RHS.
@@ -328,10 +283,9 @@ namespace yy {
   };
 
 } // yy
-
-/* Line 35 of lalr1.cc  */
-#line 334 "knot_parser/knot_parser.hh"
+/* Line 36 of lalr1.cc  */
+#line 288 "knot_parser/knot_parser.hh"
 
 
 
-#endif /* ! defined PARSER_HEADER_H */
+#endif /* !YY_YY_KNOT_PARSER_KNOT_PARSER_HH_INCLUDED  */
diff --git a/knot_parser/knot_parser.yy b/knot_parser/knot_parser.yy
index 9210974..2a30bdc 100644
--- a/knot_parser/knot_parser.yy
+++ b/knot_parser/knot_parser.yy
@@ -24,6 +24,7 @@ YY_DECL;
   basedvector<int, 1> *int_vec;
   basedvector<basedvector<int, 1>, 1> *int_vec2;
   const char *string;
+  knot_diagram *kd;
 }
 
 %code {
@@ -36,9 +37,12 @@ YY_DECL;
 
 %token PD DT BR LINK X T
 %token U UNKNOT
+
 %token <alternating> ALT NONALT
 %token <string> STRING
 
+%type <kd> knot knot_1
+%type <kd> rolfsen_knot htw_knot mt_link planar_diagram dt torus_link unknot braid
 %type <alternating> alt_spec
 
 %type <int_vec> int_vec int_vec_1
@@ -49,9 +53,27 @@ YY_DECL;
 
 %%
 
-%start knot;
+%start entry;
+
+entry:
+    knot
+      { 
+        parsed_knot = *$1; 
+        delete $1;
+      }
+  ;
 
 knot:
+   knot_1
+ | knot knot
+     {
+       $$ = new knot_diagram (DISJOINT_UNION, *$1, *$2);
+       delete $1;
+       delete $2;
+     }
+ ;
+
+knot_1:
     rolfsen_knot
   | htw_knot
   | mt_link
@@ -70,7 +92,7 @@ rolfsen_knot:
 	
 	if (n >= 1 && n <= 10
 	    && k >= 1 && k <= rolfsen_crossing_knots (n))
-	  parsed_knot = knot_diagram (rolfsen_knot (n, k));
+	  $$ = new knot_diagram (rolfsen_knot (n, k));
 	else
 	  {
 	    fprintf (stderr, "knot_parser: no such Rolfsen knot `%d_%d'\n",
@@ -89,7 +111,7 @@ htw_knot:
 	
 	if (n >= 1 && n <= 16
 	    && k >= 1 && k <= htw_knots (n, alt))
-	  parsed_knot = knot_diagram (htw_knot (n, alt, k));
+	  $$ = new knot_diagram (htw_knot (n, alt, k));
 	else
 	  {
 	    fprintf (stderr, "knot_parser: no such HTW knot `%d%c%d'\n",
@@ -108,7 +130,7 @@ mt_link:
 	
 	if (n >= 1 && n <= 14
 	    && k >= 1 && k <= mt_links (n, alt))
-	  parsed_knot = knot_diagram (mt_link (n, alt, k));
+	  $$ = new knot_diagram (mt_link (n, alt, k));
 	else
 	  {
 	    fprintf (stderr, "knot_parser: no such MT link `%d%c%d'\n", 
@@ -120,9 +142,9 @@ mt_link:
 
 planar_diagram:
     PD '[' int_vec2 ']'
-      { parsed_knot = knot_diagram (planar_diagram ("<parsed>", *$3)); }
+      { $$ = new knot_diagram (planar_diagram ("<parsed>", *$3)); }
   | PD '[' crossing_vec ']'
-      { parsed_knot = knot_diagram (planar_diagram ("<parsed>", *$3)); }
+      { $$ = new knot_diagram (planar_diagram ("<parsed>", *$3)); }
   ;
 
 dt:
@@ -130,22 +152,22 @@ dt:
       {
 	basedvector<basedvector<int, 1>, 1> even_labels (1);
 	even_labels[1] = *$3;
-	parsed_knot = knot_diagram (dt_code ("<parsed>", even_labels));
+	$$ = new knot_diagram (dt_code ("<parsed>", even_labels));
       }
   | DT '[' int_vec2 ']'
-      { parsed_knot = knot_diagram (dt_code ("<parsed>", *$3)); }
+      { $$ = new knot_diagram (dt_code ("<parsed>", *$3)); }
   | DT '[' STRING ']'
-      { parsed_knot = knot_diagram (dt_code ("<parsed>", $3)); }
+      { $$ = new knot_diagram (dt_code ("<parsed>", $3)); }
   ;
 
 torus_link:
     T '(' INT ',' INT ')'
-      { parsed_knot = knot_diagram (torus_knot ($3, $5)); }
+      { $$ = new knot_diagram (torus_knot ($3, $5)); }
   ;
 
 braid:
     BR '[' INT ',' int_vec ']'
-      { parsed_knot = knot_diagram (braid ($3, *$5)); }
+      { $$ = new knot_diagram (braid ($3, *$5)); }
   ;
 
 unknot:
@@ -154,7 +176,7 @@ unknot:
 	unsigned unknot_ar[1][4] = {
 	  { 2, 1, 3, 4, },
 	};
-	parsed_knot = knot_diagram ("U", 1, unknot_ar);
+	$$ = new knot_diagram ("U", 1, unknot_ar);
       }
   ;
 
diff --git a/knot_parser/location.hh b/knot_parser/location.hh
index c123693..7faaf2b 100644
--- a/knot_parser/location.hh
+++ b/knot_parser/location.hh
@@ -1,9 +1,8 @@
-/* A Bison parser, made by GNU Bison 2.4.3.  */
+/* A Bison parser, made by GNU Bison 2.6.5.  */
 
 /* Locations for Bison parsers in C++
    
-      Copyright (C) 2002, 2003, 2004, 2005, 2006, 2007, 2009, 2010 Free Software
-   Foundation, Inc.
+      Copyright (C) 2002-2007, 2009-2012 Free Software Foundation, Inc.
    
    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
@@ -32,39 +31,55 @@
    version 2.2 of Bison.  */
 
 /**
- ** \file location.hh
+ ** \file knot_parser/location.hh
  ** Define the yy::location class.
  */
 
-#ifndef BISON_LOCATION_HH
-# define BISON_LOCATION_HH
+#ifndef YY_YY_KNOT_PARSER_LOCATION_HH_INCLUDED
+# define YY_YY_KNOT_PARSER_LOCATION_HH_INCLUDED
 
-# include <iostream>
-# include <string>
 # include "position.hh"
 
 
 namespace yy {
-
-/* Line 163 of location.cc  */
-#line 51 "knot_parser/location.hh"
+/* Line 164 of location.cc  */
+#line 47 "knot_parser/location.hh"
 
   /// Abstract a location.
   class location
   {
   public:
 
-    /// Construct a location.
-    location ()
-      : begin (), end ()
+    /// Construct a location from \a b to \a e.
+    location (const position& b, const position& e)
+      : begin (b)
+      , end (e)
+    {
+    }
+
+    /// Construct a 0-width location in \a p.
+    explicit location (const position& p = position ())
+      : begin (p)
+      , end (p)
+    {
+    }
+
+    /// Construct a 0-width location in \a f, \a l, \a c.
+    explicit location (std::string* f,
+                       unsigned int l = 1u,
+                       unsigned int c = 1u)
+      : begin (f, l, c)
+      , end (f, l, c)
     {
     }
 
 
     /// Initialization.
-    inline void initialize (std::string* fn)
+    void initialize (std::string* f = YY_NULL,
+                     unsigned int l = 1u,
+                     unsigned int c = 1u)
     {
-      begin.initialize (fn);
+      begin.initialize (f, l, c);
       end = begin;
     }
 
@@ -72,19 +87,19 @@ namespace yy {
      ** \{ */
   public:
     /// Reset initial location to final location.
-    inline void step ()
+    void step ()
     {
       begin = end;
     }
 
     /// Extend the current location to the COUNT next columns.
-    inline void columns (unsigned int count = 1)
+    void columns (unsigned int count = 1)
     {
       end += count;
     }
 
     /// Extend the current location to the COUNT next lines.
-    inline void lines (unsigned int count = 1)
+    void lines (unsigned int count = 1)
     {
       end.lines (count);
     }
@@ -158,8 +173,7 @@ namespace yy {
 
 
 } // yy
+/* Line 292 of location.cc  */
+#line 178 "knot_parser/location.hh"
 
-/* Line 272 of location.cc  */
-#line 164 "knot_parser/location.hh"
-
-#endif // not BISON_LOCATION_HH
+#endif /* !YY_YY_KNOT_PARSER_LOCATION_HH_INCLUDED  */
diff --git a/knot_parser/position.hh b/knot_parser/position.hh
index c034a2e..dbbea36 100644
--- a/knot_parser/position.hh
+++ b/knot_parser/position.hh
@@ -1,9 +1,8 @@
-/* A Bison parser, made by GNU Bison 2.4.3.  */
+/* A Bison parser, made by GNU Bison 2.6.5.  */
 
 /* Positions for Bison parsers in C++
    
-      Copyright (C) 2002, 2003, 2004, 2005, 2006, 2007, 2009, 2010 Free Software
-   Foundation, Inc.
+      Copyright (C) 2002-2007, 2009-2012 Free Software Foundation, Inc.
    
    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
@@ -32,60 +31,71 @@
    version 2.2 of Bison.  */
 
 /**
- ** \file position.hh
+ ** \file knot_parser/position.hh
  ** Define the yy::position class.
  */
 
-#ifndef BISON_POSITION_HH
-# define BISON_POSITION_HH
+#ifndef YY_YY_KNOT_PARSER_POSITION_HH_INCLUDED
+# define YY_YY_KNOT_PARSER_POSITION_HH_INCLUDED
 
+# include <algorithm> // std::max
 # include <iostream>
 # include <string>
-# include <algorithm>
+
+# ifndef YY_NULL
+#  if defined __cplusplus && 201103L <= __cplusplus
+#   define YY_NULL nullptr
+#  else
+#   define YY_NULL 0
+#  endif
+# endif
 
 
 namespace yy {
-
 /* Line 38 of location.cc  */
-#line 51 "knot_parser/position.hh"
+#line 57 "knot_parser/position.hh"
   /// Abstract a position.
   class position
   {
   public:
 
     /// Construct a position.
-    position ()
-      : filename (0), line (1), column (1)
+    explicit position (std::string* f = YY_NULL,
+                       unsigned int l = 1u,
+                       unsigned int c = 1u)
+      : filename (f)
+      , line (l)
+      , column (c)
     {
     }
 
 
     /// Initialization.
-    inline void initialize (std::string* fn)
+    void initialize (std::string* fn = YY_NULL,
+                     unsigned int l = 1u,
+                     unsigned int c = 1u)
     {
       filename = fn;
-      line = 1;
-      column = 1;
+      line = l;
+      column = c;
     }
 
     /** \name Line and Column related manipulators
      ** \{ */
-  public:
     /// (line related) Advance to the COUNT next lines.
-    inline void lines (int count = 1)
+    void lines (int count = 1)
     {
-      column = 1;
+      column = 1u;
       line += count;
     }
 
     /// (column related) Advance to the COUNT next columns.
-    inline void columns (int count = 1)
+    void columns (int count = 1)
     {
       column = std::max (1u, column + count);
     }
     /** \} */
 
-  public:
     /// File name to which this position refers.
     std::string* filename;
     /// Current line number.
@@ -95,7 +105,7 @@ namespace yy {
   };
 
   /// Add and assign a position.
-  inline const position&
+  inline position&
   operator+= (position& res, const int width)
   {
     res.columns (width);
@@ -111,7 +121,7 @@ namespace yy {
   }
 
   /// Add and assign a position.
-  inline const position&
+  inline position&
   operator-= (position& res, const int width)
   {
     return res += -width;
@@ -156,7 +166,6 @@ namespace yy {
 
 
 } // yy
-
-/* Line 145 of location.cc  */
-#line 162 "knot_parser/position.hh"
-#endif // not BISON_POSITION_HH
+/* Line 149 of location.cc  */
+#line 171 "knot_parser/position.hh"
+#endif /* !YY_YY_KNOT_PARSER_POSITION_HH_INCLUDED  */
diff --git a/knot_parser/stack.hh b/knot_parser/stack.hh
index 3c16aa6..e005e60 100644
--- a/knot_parser/stack.hh
+++ b/knot_parser/stack.hh
@@ -1,9 +1,8 @@
-/* A Bison parser, made by GNU Bison 2.4.3.  */
+/* A Bison parser, made by GNU Bison 2.6.5.  */
 
 /* Stack handling for Bison parsers in C++
    
-      Copyright (C) 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-   Software Foundation, Inc.
+      Copyright (C) 2002-2012 Free Software Foundation, Inc.
    
    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
@@ -31,21 +30,24 @@
    This special exception was added by the Free Software Foundation in
    version 2.2 of Bison.  */
 
-#ifndef BISON_STACK_HH
-# define BISON_STACK_HH
+/**
+ ** \file knot_parser/stack.hh
+ ** Define the yy::stack class.
+ */
 
-#include <deque>
+#ifndef YY_YY_KNOT_PARSER_STACK_HH_INCLUDED
+# define YY_YY_KNOT_PARSER_STACK_HH_INCLUDED
+
+# include <deque>
 
 
 namespace yy {
-
-/* Line 1067 of lalr1.cc  */
-#line 44 "knot_parser/stack.hh"
+/* Line 37 of stack.hh  */
+#line 47 "knot_parser/stack.hh"
   template <class T, class S = std::deque<T> >
   class stack
   {
   public:
-
     // Hide our reversed order.
     typedef typename S::reverse_iterator iterator;
     typedef typename S::const_reverse_iterator const_iterator;
@@ -84,7 +86,7 @@ namespace yy {
     pop (unsigned int n = 1)
     {
       for (; n; --n)
-	seq_.pop_front ();
+        seq_.pop_front ();
     }
 
     inline
@@ -98,7 +100,6 @@ namespace yy {
     inline const_iterator end () const { return seq_.rend (); }
 
   private:
-
     S seq_;
   };
 
@@ -107,10 +108,9 @@ namespace yy {
   class slice
   {
   public:
-
-    slice (const S& stack,
-	   unsigned int range) : stack_ (stack),
-				 range_ (range)
+    slice (const S& stack, unsigned int range)
+      : stack_ (stack)
+      , range_ (range)
     {
     }
 
@@ -122,15 +122,12 @@ namespace yy {
     }
 
   private:
-
     const S& stack_;
     unsigned int range_;
   };
 
 } // yy
+/* Line 119 of stack.hh  */
+#line 132 "knot_parser/stack.hh"
 
-/* Line 1153 of lalr1.cc  */
-#line 134 "knot_parser/stack.hh"
-
-#endif // not BISON_STACK_HH[]dnl
-
+#endif /* !YY_YY_KNOT_PARSER_STACK_HH_INCLUDED  */
diff --git a/knot_tables.cpp b/knot_tables.cpp
index 2acaae2..3733799 100644
--- a/knot_tables.cpp
+++ b/knot_tables.cpp
@@ -402,7 +402,7 @@ htw_knot (unsigned n, bool alternating, unsigned k)
 	before += htw_alternating[i];
       off = 8 * (before + k - 1);
       
-      file = HOME "alternating";
+      file = HOME "/alternating";
     }
   else
     {
@@ -411,7 +411,7 @@ htw_knot (unsigned n, bool alternating, unsigned k)
 	before += htw_nonalternating[i];
       off = 10 * (before + k - 1);
       
-      file = HOME "nonalternating";
+      file = HOME "/nonalternating";
     }
   FILE *fp = fopen (file, "r");
   if (fp == 0)
@@ -499,6 +499,13 @@ mt_links (unsigned n, bool alternating)
     return mt_nonalternating[n - 1];
 }
 
+unsigned
+mt_links (unsigned n)
+{
+  assert (between (1, n, 14));
+  return mt_alternating[n - 1] + mt_nonalternating[n - 1];
+}
+
 dt_code
 mt_link (unsigned n, bool alternating, unsigned k)
 {
@@ -507,7 +514,7 @@ mt_link (unsigned n, bool alternating, unsigned k)
   
   char buf[1000];
   
-  sprintf (buf, HOME "mtlinks/hyperbolic_data_%02d%c", n, alternating ? 'a' : 'n');
+  sprintf (buf, HOME "/mtlinks/hyperbolic_data_%02d%c", n, alternating ? 'a' : 'n');
   FILE *fp = fopen (buf, "r");
   if (fp == 0)
     {
@@ -535,6 +542,19 @@ mt_link (unsigned n, bool alternating, unsigned k)
   return dt_code (buf2, buf);
 }
 
+dt_code
+mt_link (unsigned n, unsigned k)
+{
+  assert (between (1, n, 16));
+  assert (k >= 1);
+  
+  unsigned na = mt_links (n, 1);
+  if (k <= na)
+    return mt_link (n, 1, k);
+  else
+    return mt_link (n, 0, k - na);
+}
+
 planar_diagram
 torus_knot (unsigned n_strands, unsigned n_shifts)
 {
@@ -594,7 +614,7 @@ torus_knot (unsigned n_strands, unsigned n_shifts)
   return planar_diagram (std::string (buf), crossings);
 }
 
-planar_diagram
+knot_diagram
 braid (unsigned n_strands, unsigned n_twists, int twists_ar[])
 {
   basedvector<int, 1> twists (n_twists);
@@ -603,10 +623,9 @@ braid (unsigned n_strands, unsigned n_twists, int twists_ar[])
   return braid (n_strands, twists);
 }
 
-planar_diagram
+knot_diagram
 braid (unsigned n_strands, const basedvector<int, 1> &twists)
 {
-  unsigned n_crossings = twists.size ();
   unsigned e = 0;
   
   basedvector<unsigned, 1> final_strands (n_strands);
@@ -622,22 +641,44 @@ braid (unsigned n_strands, const basedvector<int, 1> &twists)
     last_twist[i] = 0;
   for (unsigned i = 1; i <= twists.size (); i ++)
     {
-      unsigned t = std::abs (twists[i]);
+      unsigned t = abs (twists[i]);
       last_twist[t] = i;
       last_twist[t + 1] = i;
     }
-#ifndef NDEBUG
-  for (unsigned i = 1; i <= n_strands; i ++)
-    assert (last_twist[i] != 0);
-#endif
   
-  basedvector<basedvector<int, 1>, 1> crossings (n_crossings);
+  unsigned n_crossings = twists.size ();
+  for (unsigned i = 1; i <= n_strands; i ++)
+    {
+      if (last_twist[i] == 0)
+	n_crossings ++;
+    }
+  
+  basedvector<basedvector<unsigned, 1>, 1> crossings (n_crossings);
   for (unsigned i = 1; i <= n_crossings; i ++)
-    crossings[i] = basedvector<int, 1> (4);
+    crossings[i] = basedvector<unsigned, 1> (4);
+  
+  unsigned c = twists.size ();
+  for (unsigned i = 1; i <= n_strands; i ++)
+    {
+      if (last_twist[i] == 0)
+	{
+	  ++ c;
+	  unsigned e1 = strands[i];
+	  unsigned e2 = ++ e;
+	  
+	  strands[i] = final_strands[i] = 0;
+	  
+	  crossings[c][1] = edge_from_ept (e1);
+	  crossings[c][2] = edge_to_ept (e2);
+	  crossings[c][3] = edge_from_ept (e2);
+	  crossings[c][4] = edge_to_ept (e1);
+	}
+    }
+  assert (c == n_crossings);
   
   for (unsigned i = 1; i <= twists.size (); i ++)
     {
-      unsigned t = std::abs (twists[i]);
+      unsigned t = abs (twists[i]);
       unsigned e1 = strands[t],
 	e4 = strands[t + 1];
       unsigned e2, e3;
@@ -668,27 +709,30 @@ braid (unsigned n_strands, const basedvector<int, 1> &twists)
       
       if (twists[i] > 0)
 	{
-	  crossings[i][1] = e1;
-	  crossings[i][2] = e2;
-	  crossings[i][3] = e3;
-	  crossings[i][4] = e4;
+	  crossings[i][1] = edge_to_ept (e1);
+	  crossings[i][2] = edge_from_ept (e2);
+	  crossings[i][3] = edge_from_ept (e3);
+	  crossings[i][4] = edge_to_ept (e4);
 	}
       else
 	{
-	  crossings[i][1] = e2;
-	  crossings[i][2] = e3;
-	  crossings[i][3] = e4;
-	  crossings[i][4] = e1;
+	  crossings[i][1] = edge_from_ept (e2);
+	  crossings[i][2] = edge_from_ept (e3);
+	  crossings[i][3] = edge_to_ept (e4);
+	  crossings[i][4] = edge_to_ept (e1);
 	}
     }
   
   assert (e == n_crossings * 2);
 #ifndef NDEBUG
   for (unsigned i = 1; i <= n_strands; i ++)
-    assert (final_strands[i] == 0);
+    {
+      assert (strands[i] == 0);
+      assert (final_strands[i] == 0);
+    }
 #endif
   
-  return planar_diagram ("abraid", crossings);
+  return knot_diagram ("abraid", crossings);
 }
 
 basedvector<basedvector<unsigned, 1>, 1> 
@@ -697,7 +741,7 @@ mutant_knot_groups (unsigned n)
   assert (11 <= n && n <= 15);
   
   char buf[1000];
-  sprintf (buf, HOME "mutant_knot_groups/dat%d", n);
+  sprintf (buf, HOME "/mutant_knot_groups/dat%d", n);
   
   FILE *fp = fopen (buf, "r");
   if (fp == 0)
@@ -728,3 +772,138 @@ mutant_knot_groups (unsigned n)
   
   return r;
 }
+
+knot_diagram
+knot_desc::diagram () const
+{
+  switch (t)
+    {
+    case ROLFSEN:
+      return knot_diagram (rolfsen_knot (i, j));
+      
+    case HTW:
+      return knot_diagram (htw_knot (i, j));
+    case HTW_ALT:
+      return knot_diagram (htw_knot (i, 1, j));
+    case HTW_NONALT:
+      return knot_diagram (htw_knot (i, 0, j));
+
+    case MT:
+      return knot_diagram (mt_link (i, j));
+    case MT_ALT:
+      return knot_diagram (mt_link (i, 1, j));
+    case MT_NONALT:
+      return knot_diagram (mt_link (i, 0, j));
+      
+    case TORUS:
+      return knot_diagram (torus_knot (i, j));
+      
+    default: abort ();
+    }
+}
+  
+std::string
+knot_desc::name () const
+{
+  char buf[1000];
+
+#if 0
+  sprintf (buf, "knot_desc(%d, %d, %d)", (int)t, i, j);
+  return buf;
+#endif
+  
+  switch (t)
+    {
+    case ROLFSEN:
+      sprintf (buf, "%d_%d", i, j);
+      break;
+      
+    case HTW:
+      {
+	unsigned na = htw_knots (i, 1);
+	if (j <= na)
+	  sprintf (buf, "%da%d", i, j);
+	else
+	  sprintf (buf, "%dn%d", i, j - na);
+      }
+      break;
+      
+    case HTW_ALT:
+      sprintf (buf, "%da%d", i, j);
+      break;
+
+    case HTW_NONALT:
+      sprintf (buf, "%dn%d", i, j);
+      break;
+
+    case MT:
+      {
+	unsigned na = mt_links (i, 1);
+	if (j <= na)
+	  sprintf (buf, "L%da%d", i, j);
+	else
+	  sprintf (buf, "L%dn%d", i, j - na);
+      }
+      break;
+      
+    case MT_ALT:
+      sprintf (buf, "L%da%d", i, j);
+      break;
+
+    case MT_NONALT:
+      sprintf (buf, "L%dn%d", i, j);
+      break;
+      
+    case TORUS:
+      sprintf (buf, "T(%d, %d)", i, j);
+      break;
+      
+    default: abort ();
+    }
+  
+  return buf;
+}
+
+unsigned
+knot_desc::table_crossing_knots () const
+{
+  switch (t)
+    {
+    case ROLFSEN:
+      return rolfsen_crossing_knots (i);
+      
+    case HTW:
+      return htw_knots (i);
+    case HTW_ALT:
+      return htw_knots (i, 1);
+    case HTW_NONALT:
+      return htw_knots (i, 0);
+
+    case MT:
+      return mt_links (i);
+    case MT_ALT:
+      return mt_links (i, 1);
+    case MT_NONALT:
+      return mt_links (i, 0);
+      
+    default: abort ();
+    }
+}
+
+knot_desc::knot_desc (reader &r)
+{
+  int x;
+  read (r, x);
+  t = (table)x;
+  
+  read (r, i);
+  read (r, j);
+}
+
+void
+knot_desc::write_self (writer &w) const
+{
+  write (w, (int)t);
+  write (w, i);
+  write (w, j);
+}
diff --git a/knotkit.h b/knotkit.h
index c0165ae..b1974e6 100644
--- a/knotkit.h
+++ b/knotkit.h
@@ -14,13 +14,90 @@ class knot_diagram;
 #include <dt_code.h>
 #include <knot_diagram.h>
 
+#include <simplify_chain_complex.h>
 #include <sseq.h>
 #include <smoothing.h>
 #include <cobordism.h>
 
 #include <cube.h>
+#include <steenrod_square.h>
 #include <spanning_tree_complex.h>
 
+class knot_desc
+{
+ public:
+  enum table
+  {
+    NONE,
+    ROLFSEN,
+    HTW,
+    HTW_ALT,
+    HTW_NONALT,
+    MT,
+    MT_ALT,
+    MT_NONALT,
+    TORUS,
+  };
+  
+  table t;
+  unsigned i;
+  unsigned j;
+  
+ public:
+  knot_desc () : t(NONE) { }
+  knot_desc (table t_, unsigned i_, unsigned j_)
+    : t(t_), i(i_), j(j_)
+  { }
+  knot_desc (const knot_desc &desc)
+    : t(desc.t), i(desc.i), j(desc.j)
+  { }
+  knot_desc (reader &r);
+  ~knot_desc () { }
+  
+  knot_desc &operator = (const knot_desc &desc)
+  {
+    t = desc.t;
+    i = desc.i;
+    j = desc.j;
+    return *this;
+  }
+  
+  bool operator == (const knot_desc &desc) const
+  {
+    return t == desc.t && i == desc.i && j == desc.j;
+  }
+  bool operator < (const knot_desc &desc) const
+  {
+    if ((int)t < (int)desc.t)
+      return 1;
+    else if ((int)t > (int)desc.t)
+      return 0;
+    
+    if (i < desc.i)
+      return 1;
+    else if (i > desc.i)
+      return 0;
+    
+    return j < desc.j;
+  }
+  
+  knot_diagram diagram () const;
+  
+  std::string name () const;
+  unsigned table_crossing_knots () const;
+  
+  hash_t hash_self () const
+  {
+    return hash_combine (hash ((int)t),
+			 hash_combine (hash (i),
+				       hash (j)));
+  }
+  
+  void write_self (writer &w) const;
+  void show_self () const { printf ("%s", name ().c_str ()); }
+  void display_self () const { show_self (); newline (); }
+};
+
 unsigned rolfsen_crossing_knots (unsigned n);
 planar_diagram rolfsen_knot (unsigned n, unsigned k);
 
@@ -31,12 +108,15 @@ dt_code htw_knot (unsigned n, bool alternating, unsigned k);
 dt_code htw_knot (unsigned n, unsigned k);
 
 unsigned mt_links (unsigned n, bool alternating);
+unsigned mt_links (unsigned n);
+
 dt_code mt_link (unsigned n, bool alternating, unsigned k);
+dt_code mt_link (unsigned n, unsigned k);
 
 planar_diagram torus_knot (unsigned n_strands, unsigned n_shifts);
 
-planar_diagram braid (unsigned n_strands, const basedvector<int, 1> &twists);
-planar_diagram braid (unsigned n_strands, unsigned n_twists, int twists_ar[]);
+knot_diagram braid (unsigned n_strands, const basedvector<int, 1> &twists);
+knot_diagram braid (unsigned n_strands, unsigned n_twists, int twists_ar[]);
 
 knot_diagram parse_knot (const char *s);
 resolution_diagram parse_resolution_diagram (const char *s);
diff --git a/lib/hashmap.h b/lib/hashmap.h
index 1616840..75ddae2 100644
--- a/lib/hashmap.h
+++ b/lib/hashmap.h
@@ -1,23 +1,29 @@
 
 /* wrapper for std::unordered_map */
 
-template<class K>
-struct hasher : public std::unary_function<K, hash_t>
-{
-  hash_t operator () (const K &k) const { return hash (k); }
-};
-
 template<class K, class V>
-class hashmap : public map_wrapper<std::tr1::unordered_map<K, V, hasher<K> >, K, V>
+class hashmap : public map_wrapper<std::unordered_map<K, V, hasher<K> >, K, V>
 {
-  typedef map_wrapper<std::tr1::unordered_map<K, V, hasher<K> >, K, V> base;
+  typedef map_wrapper<std::unordered_map<K, V, hasher<K> >, K, V> base;
   
  public:
   hashmap () { }
   hashmap (const hashmap &m) : base(m) { }
   hashmap (copy, const hashmap &m) : base(COPY, m) { }
+  hashmap (initializer_list<std::pair<const K, V> > il) : base(il) { }
   hashmap (reader &r) : base(r) { }
   ~hashmap () { }
   
   hashmap &operator = (const hashmap &m) { base::operator = (m); return *this; }
+  hashmap &operator = (initializer_list<std::pair<const K, V> > il)
+  {
+    base::operator = (il);
+    return *this;
+  }
 };
+
+template<class K, class V>
+using hashmap_iter = map_wrapper_iter<std::unordered_map<K, V>, K, V>;
+
+template<class K, class V>
+using hashmap_const_iter = map_wrapper_const_iter<std::unordered_map<K, V>, K, V>;
diff --git a/lib/hashset.h b/lib/hashset.h
new file mode 100644
index 0000000..971eb2a
--- /dev/null
+++ b/lib/hashset.h
@@ -0,0 +1,25 @@
+
+/* wrapper for std::unordered_set */
+
+template<class T>
+class hashset : public set_wrapper<std::unordered_set<T, hasher<T> >, T>
+{
+  typedef set_wrapper<std::unordered_set<T, hasher<T> >, T> base;
+  
+ public:
+  hashset () { }
+  hashset (const hashset &m) : base(m) { }
+  hashset (copy, const hashset &m) : base(COPY, m) { }
+  hashset (initializer_list<T> il) : base(il) { }
+  hashset (reader &r) : base(r) { }
+  ~hashset () { }
+  
+  hashset &operator = (const hashset &m) { base::operator = (m); return *this; }
+  hashset &operator = (initializer_list<T> il) { base::operator = (il); return *this; }
+};
+
+template<class T>
+using hashset_iter = set_wrapper_iter<std::unordered_set<T, hasher<T> >, T>;
+
+template<class T>
+using hashset_const_iter = set_wrapper_const_iter<std::unordered_set<T, hasher<T> >, T>;
diff --git a/lib/io.cpp b/lib/io.cpp
index a02ccd0..c6a75fc 100644
--- a/lib/io.cpp
+++ b/lib/io.cpp
@@ -1,10 +1,236 @@
 
-#include <lib/lib.h>
+// #include <lib/lib.h>
+#include <algebra/algebra.h>
+
+writer::writer (bool raw_)
+  : raw(raw_),
+    aw(new algebra_writer)
+{
+}
+
+writer::~writer ()
+{
+  delete aw;
+}
+
+void
+writer::write_int (int x)
+{
+  if (raw)
+    write_raw (&x, sizeof x, 1);
+  else
+    {
+      uint8 buf[5];
+      unsigned n = 0;
+      
+      bool more = 1;
+      while (more)
+	{
+	  uint8 b = (uint8)(x & 0x7f);
+	  x >>= 7;
+      
+	  if ((x == 0
+	       && ! (b & 0x40))
+	      || (x == -1
+		  && (b & 0x40) == 0x40))
+	    more = 0;
+	  else
+	    b |= 0x80;
+      
+	  assert (n < 5);
+	  buf[n] = b;
+	  n ++;
+	}
+      
+      write_raw (buf, sizeof buf[0], n);
+    }
+}
+
+void
+writer::write_unsigned (unsigned x)
+{
+  if (raw)
+    write_raw (&x, sizeof x, 1);
+  else
+    {
+      uint8 buf[5];
+      unsigned n = 0;
+  
+      bool more = 1;
+      while (more)
+	{
+	  uint8 b = (uint8)(x & 0x7f);
+	  x >>= 7;
+      
+	  if ((x == 0
+	       && ! (b & 0x40)))
+	    more = 0;
+	  else
+	    b |= 0x80;
+      
+	  assert (n < 5);
+	  buf[n] = b;
+	  n ++;
+	}
+  
+      write_raw (buf, sizeof buf[0], n);
+    }
+}
+
+void
+writer::write_uint64 (uint64 x)
+{
+  if (raw)
+    write_raw (&x, sizeof x, 1);
+  else
+    {
+      uint8 buf[10];
+      unsigned n = 0;
+  
+      bool more = 1;
+      while (more)
+	{
+	  uint8 b = (uint8)(x & 0x7f);
+	  x >>= 7;
+      
+	  if ((x == 0
+	       && ! (b & 0x40)))
+	    more = 0;
+	  else
+	    b |= 0x80;
+
+	  assert (n < 10);
+	  buf[n] = b;
+	  n ++;
+	}
+  
+      write_raw (buf, sizeof buf[0], n);
+    }
+}
+
+void
+writer::write_mpz (const mpz_t x)
+{
+  assert (!raw);
+  
+  size_t count;
+  void *buf = mpz_export (nullptr, &count, -1, 1, -1, 0, x);
+  
+  write_unsigned ((unsigned)count);
+  write_raw (buf, 1, count);
+  
+  free (buf);
+}
+
+reader::reader (bool raw_)
+  : raw(raw_),
+    ar(new algebra_reader)
+{
+}
+
+reader::~reader ()
+{
+  delete ar;
+}
+
+int
+reader::read_int ()
+{
+  int x = 0;
+  
+  if (raw)
+    read_raw (&x, sizeof x, 1);
+  else
+    {
+      int shift = 0;
+      for (;;)
+	{
+	  uint8 b = read_uint8 ();
+	  x |= ((int)(b & 0x7f) << shift);
+	  shift += 7;
+	  if (! (b & 0x80))
+	    {
+	      if (shift < int_bits
+		  && (b & 0x40))
+		x = (x << (int_bits - shift)) >> (int_bits - shift);
+	      break;
+	    }
+	}
+    }
+  
+  return x;
+}
+
+unsigned
+reader::read_unsigned ()
+{
+  unsigned x = 0;
+  
+  if (raw)
+    read_raw (&x, sizeof x, 1);
+  else
+    {
+      unsigned shift = 0;
+      for (;;)
+	{
+	  uint8 b = read_uint8 ();
+	  x |= ((unsigned)(b & 0x7f) << shift);
+	  shift += 7;
+	  if (! (b & 0x80))
+	    break;
+	}
+    }
+  
+  return x;
+}
+
+uint64
+reader::read_uint64 ()
+{
+  uint64 x = 0;
+
+  if (raw)
+    read_raw (&x, sizeof x, 1);
+  else
+    {
+      uint64 shift = 0;
+      for (;;)
+	{
+	  uint8 b = read_uint8 ();
+	  x |= ((uint64)(b & 0x7f) << shift);
+	  shift += 7;
+	  if (! (b & 0x80))
+	    break;
+	}
+    }
+  
+  return x;
+}
+
+void
+reader::read_mpz (mpz_t x)
+{
+  assert (!raw);
+  
+  unsigned count = read_unsigned ();
+  void *p = malloc (count);
+  if (!p)
+    {
+      stderror ("malloc");
+      exit (EXIT_FAILURE);
+    }
+  
+  read_raw (p, 1, count);
+  
+  mpz_import (x, count, -1, 1, -1, 0, p);
+  
+  free (p);
+}
 
 FILE *open_file (const std::string &file, const char *mode)
 {
   FILE *fp = fopen (file.c_str (), mode);
-  if (fp == 0)
+  if (!fp)
     {
       stderror ("fopen: %s", file.c_str ());
       exit (EXIT_FAILURE);
@@ -17,41 +243,32 @@ void close_file (FILE *fp)
   fclose (fp);
 }
 
-
-writer::writer (const std::string &file)
-  : fp(0)
+gzFile
+open_gzfile (const std::string &file, const char *mode)
 {
-  fp = open_file (file, "w");
-}
-
-writer::~writer ()
-{
-  if (fp)
+  gzFile gzfp = gzopen (file.c_str (), mode);
+  if (!gzfp)
     {
-      fclose (fp);
-      fp = 0;
+      stderror ("gzopen: %s", file.c_str ());
+      exit (EXIT_FAILURE);
     }
+  return gzfp;
 }
 
-reader::reader (const std::string &file)
-  : fp(0)
+extern void close_gzfile (gzFile gzfp)
 {
-  fp = open_file (file, "r");
-}
-
-reader::~reader ()
-{
-  if (fp)
+  int r = gzclose (gzfp);
+  if (r != Z_OK)
     {
-      fclose (fp);
-      fp = 0;
+      stderror ("gzclose");
+      exit (EXIT_FAILURE);
     }
 }
 
 void
-writer::write_char (char x)
+file_writer::write_raw (const void *p, size_t itemsize, size_t nitems)
 {
-  if (fwrite (&x, sizeof x, 1, fp) != 1)
+  if (fwrite (p, itemsize, nitems, fp) != nitems)
     {
       stderror ("fwrite");
       exit (EXIT_FAILURE);
@@ -59,103 +276,47 @@ writer::write_char (char x)
 }
 
 void
-writer::write_bool (bool x)
+file_writer::write_mpz (const mpz_t x)
 {
-  if (fwrite (&x, sizeof x, 1, fp) != 1)
+  mpz_out_raw (fp, x);
+}
+
+void
+file_reader::read_raw (void *p, size_t itemsize, size_t nitems)
+{
+  if (fread (p, itemsize, nitems, fp) != nitems)
     {
-      stderror ("fwrite");
+      stderror ("fread");
       exit (EXIT_FAILURE);
     }
 }
 
 void
-writer::write_int (int x)
+file_reader::read_mpz (mpz_t x)
 {
-  if (fwrite (&x, sizeof x, 1, fp) != 1)
+  mpz_inp_raw (x, fp);
+}
+
+void
+gzfile_writer::write_raw (const void *p, size_t itemsize, size_t nitems)
+{
+  unsigned nbytes = itemsize*nitems;
+  if (gzwrite (gzfp, p, nbytes) != nbytes)
     {
-      stderror ("fwrite");
+      stderror ("gzwrite");
       exit (EXIT_FAILURE);
     }
 }
 
 void
-writer::write_unsigned (unsigned x)
+gzfile_reader::read_raw (void *p, size_t itemsize, size_t nitems)
 {
-  if (fwrite (&x, sizeof x, 1, fp) != 1)
-    {
-      stderror ("fwrite");
-      exit (EXIT_FAILURE);
-    }
-}
-
-void
-writer::write_uint64 (uint64 x)
-{
-  if (fwrite (&x, sizeof x, 1, fp) != 1)
-    {
-      stderror ("fwrite");
-      exit (EXIT_FAILURE);
-    }
-}
-
-bool
-reader::read_bool ()
-{
-  bool x;
-  if (fread (&x, sizeof x, 1, fp) != 1)
+  unsigned nbytes = itemsize*nitems;
+  if (gzread (gzfp, p, nbytes) != nitems)
     {
       stderror ("fread");
       exit (EXIT_FAILURE);
     }
-  return x;
-}
-
-char
-reader::read_char ()
-{
-  char x;
-  if (fread (&x, sizeof x, 1, fp) != 1)
-    {
-      stderror ("fread");
-      exit (EXIT_FAILURE);
-    }
-  return x;
-}
-
-int
-reader::read_int ()
-{
-  int x;
-  if (fread (&x, sizeof x, 1, fp) != 1)
-    {
-      stderror ("fread");
-      exit (EXIT_FAILURE);
-    }
-  return x;
-}
-
-unsigned
-reader::read_unsigned ()
-{
-  unsigned x;
-  if (fread (&x, sizeof x, 1, fp) != 1)
-    {
-      stderror ("fread");
-      exit (EXIT_FAILURE);
-    }
-  return x;
-}
-
-uint64
-reader::read_uint64 ()
-{
-  uint64 x;
-  if (fread (&x, sizeof x, 1, fp) != 1)
-    {
-      stderror ("fread");
-      exit (EXIT_FAILURE);
-    }
-  return x;
 }
 
 void
@@ -164,12 +325,7 @@ read (reader &r, std::string &s)
   unsigned n = r.read_unsigned ();
   
   char buf[n + 1];
-  unsigned k = fread (buf, sizeof (char), n + 1, r.fp);
-  if (k != n + 1)
-    {
-      stderror ("fread");
-      exit (EXIT_FAILURE);
-    }
+  r.read_raw (buf, sizeof (char), n + 1);
   assert (buf[n] == 0);
   
   s = std::string (buf);
@@ -181,9 +337,5 @@ write (writer &w, const std::string &s)
   unsigned n = s.length ();
   w.write_unsigned (n);
   
-  if (fwrite (s.c_str (), sizeof (char), n + 1, w.fp) != n + 1)
-    {
-      stderror ("fwrite");
-      exit (EXIT_FAILURE);
-    }
+  w.write_raw (s.c_str (), sizeof (char), n + 1);
 }
diff --git a/lib/io.h b/lib/io.h
index 9f499e9..1860c28 100644
--- a/lib/io.h
+++ b/lib/io.h
@@ -1,43 +1,162 @@
 
-extern FILE *open_file (const std::string &file, const char *mode);
-extern void close_file (FILE *fp);
+template<class R> class module;
+
+class algebra_writer;
+class algebra_reader;
 
 class writer
 {
  public:
-  FILE *fp;
+  bool raw;
+  algebra_writer *aw;
   
  public:
-  writer (const std::string &file);
-  writer (const writer &); // doesn't exist
-  ~writer ();
+  writer (const writer &) = delete;
+  writer (bool raw_);
+  virtual ~writer ();
   
-  writer &operator = (const writer &); // doesn't exist
+  writer &operator = (const writer &) = delete;
   
-  void write_bool (bool x);
-  void write_char (char x);
+  virtual void write_raw (const void *p, size_t itemsize, size_t nitems) = 0;
+  
+  void write_bool (bool x) { write_raw (&x, sizeof x, 1); }
+  void write_char (char x) { write_raw (&x, sizeof x, 1); }
+  void write_uint8 (uint8 x) { write_raw (&x, sizeof x, 1); }
   void write_int (int x);
   void write_unsigned (unsigned x);
   void write_uint64 (uint64 x);
+  
+  virtual void write_mpz (const mpz_t x);
+  
+  template<class R> void write_mod (ptr<const module<R> > m);
 };
 
 class reader
 {
- public:
-  FILE *fp;
-
- public:
-  reader (const std::string &file);
-  reader (const reader &); // doesn't exist
-  ~reader ();
+  bool raw;
+  algebra_reader *ar;
   
-  reader &operator = (const reader &); // doesn't exist
+ public:
+  reader (const reader &) = delete;
+  reader (bool raw_);
+  virtual ~reader ();
+  
+  reader &operator = (const reader &) = delete;
+  
+  virtual void read_raw (void *p, size_t itemsize, size_t nitems) = 0;
+  
+  bool read_bool ()
+  {
+    bool x;
+    read_raw (&x, sizeof x, 1);
+    return x;
+  }
+  
+  char read_char ()
+  {
+    char x;
+    read_raw (&x, sizeof x, 1);
+    return x;
+  }
+  
+  uint8 read_uint8 ()
+  {
+    uint8 x;
+    read_raw (&x, sizeof x, 1);
+    return x;
+  }
   
-  bool read_bool ();
-  char read_char ();
   int read_int ();
   unsigned read_unsigned ();
   uint64 read_uint64 ();
+  
+  virtual void read_mpz (mpz_t x);
+  
+  template<class R> ptr<const module<R> > read_mod ();
+};
+
+extern FILE *open_file (const std::string &file, const char *mode);
+extern void close_file (FILE *fp);
+
+extern gzFile open_gzfile (const std::string &file, const char *mode);
+extern void close_gzfile (gzFile fp);
+
+class file_writer : public writer
+{
+ public:
+  FILE *fp;
+  
+ public:
+  file_writer (const std::string &file, bool raw = false)
+    : writer(raw),
+      fp(open_file (file, "w"))
+  {
+  }
+  file_writer (const file_writer &) = delete;
+  ~file_writer () { close_file (fp); }
+  
+  file_writer &operator = (const file_writer &) = delete;
+  
+  void write_raw (const void *p, size_t itemsize, size_t nitems);
+  void write_mpz (const mpz_t x);
+};
+
+class file_reader : public reader
+{
+ public:
+  FILE *fp;
+  
+ public:
+  file_reader (const std::string &file, bool raw = false)
+    : reader(raw),
+      fp(open_file (file, "r"))
+  {
+  }
+  file_reader (const file_reader &) = delete;
+  ~file_reader () { close_file (fp); }
+  
+  file_reader &operator = (const file_reader &) = delete;
+  
+  void read_raw (void *p, size_t itemsize, size_t nitems);
+  void read_mpz (mpz_t x);
+};
+
+class gzfile_writer : public writer
+{
+ public:
+  gzFile gzfp;
+  
+ public:
+  gzfile_writer (const std::string &file)
+    : writer(false),
+      gzfp(open_gzfile (file, "w9"))
+  {
+  }
+  gzfile_writer (const gzfile_writer &) = delete;
+  ~gzfile_writer () { close_gzfile (gzfp); }
+  
+  gzfile_writer &operator = (const gzfile_writer &) = delete;
+  
+  void write_raw (const void *p, size_t itemsize, size_t nitems);
+};
+
+class gzfile_reader : public reader
+{
+ public:
+  gzFile gzfp;
+  
+ public:
+  gzfile_reader (const std::string &file)
+    : reader(false),
+      gzfp(open_gzfile (file, "r"))
+  {
+  }
+  gzfile_reader (const gzfile_reader &) = delete;
+  ~gzfile_reader () { close_gzfile (gzfp); }
+  
+  gzfile_reader &operator = (const gzfile_reader &) = delete;
+  
+  void read_raw (void *p, size_t itemsize, size_t nitems);
 };
 
 inline void read (reader &r, bool &x) { x = r.read_bool (); }
@@ -45,7 +164,7 @@ inline void read (reader &r, char &x) { x = r.read_char (); }
 inline void read (reader &r, int &x) { x = r.read_int (); }
 inline void read (reader &r, unsigned &x) { x = r.read_unsigned (); }
 inline void read (reader &r, uint64 &x) { x = r.read_uint64 (); }
-void read (reader &r, std::string &s);
+extern void read (reader &r, std::string &s);
 template<class T> inline void read (reader &r, T &x) { x = T(r); }
 
 inline void ctor_read (reader &r, bool *p) { *p = r.read_bool (); }
@@ -60,5 +179,5 @@ inline void write (writer &w, char x) { w.write_char (x); }
 inline void write (writer &w, int x) { w.write_int (x); }
 inline void write (writer &w, unsigned x) { w.write_unsigned (x); }
 inline void write (writer &w, uint64 x) { w.write_uint64 (x); }
-void write (writer &w, const std::string &s);
+extern void write (writer &w, const std::string &s);
 template<class T> inline void write (writer &w, const T &x) { x.write_self (w); }
diff --git a/lib/lib.h b/lib/lib.h
index e72b113..1bb300c 100644
--- a/lib/lib.h
+++ b/lib/lib.h
@@ -1,5 +1,6 @@
 
 #include <assert.h>
+#include <stdlib.h>
 #include <stdio.h>
 #include <string.h>
 #include <math.h>
@@ -7,17 +8,21 @@
 #include <sys/stat.h>
 #include <sys/errno.h>
 
+#include <gmp.h>
+#include <zlib.h>
+
+#include <initializer_list>
 #include <set>
 #include <map>
 #include <string>
 #include <queue>
 #include <algorithm>
 
-#include <tr1/functional>
-#include <tr1/unordered_map>
+#include <functional>
+#include <unordered_set>
+#include <unordered_map>
 
 /* just need to implement ==, < */
-template<class T> bool operator != (const T &a, const T &b) { return ! (a == b); }
 template<class T> bool operator <= (const T &a, const T &b) { return (a < b) || (a == b); }
 template<class T> bool operator > (const T &a, const T &b) { return ! (a <= b); }
 template<class T> bool operator >= (const T &a, const T &b) { return ! (a < b); }
@@ -25,6 +30,8 @@ template<class T> bool operator >= (const T &a, const T &b) { return ! (a < b);
 typedef enum copy { COPY, COPY1 } copy;
 typedef enum copy2 { COPY2 } copy2;
 
+typedef unsigned char uint8;
+
 typedef unsigned char uchar;
 typedef unsigned short ushort;
 typedef unsigned int uint;
@@ -33,8 +40,15 @@ typedef unsigned long ulong;
 typedef long long int64;
 typedef unsigned long long uint64;
 
+inline int64 abs64 (int64 x) { return llabs (x); }
+
 typedef size_t hash_t;
 
+static const unsigned uint8_bits = 8;
+static const unsigned char_bits = 8;
+static const unsigned uchar_bits = 8;
+static const unsigned int_bits = 32;
+
 inline hash_t hash (bool x) { return (hash_t)x; }
 inline hash_t hash (char x) { return (hash_t)x; }
 inline hash_t hash (unsigned char x) { return (hash_t)x; }
@@ -147,13 +161,24 @@ void stderror (const char *fmt, ...);
 
 int alpha_to_int (char c);
 
-#include <lib/io.h>
+using std::tuple;
+using std::get;
+using std::make_tuple;
+using std::tie;
+using std::ignore;
+using std::initializer_list;
 
 #include <lib/show.h>
 #include <lib/refcount.h>
+
+#include <lib/io.h>
+
 #include <lib/pair.h>
+
 #include <lib/maybe.h>
 #include <lib/vector.h>
+
+#include <lib/set_wrapper.h>
 #include <lib/set.h>
 
 class bitset;
@@ -172,12 +197,40 @@ inline int random_int (int from, int to)
 #include <lib/bitset.h>
 #include <lib/ullmanset.h>
 #include <lib/setcommon.h>
-
 #include <lib/map_wrapper.h>
 #include <lib/map.h>
+
+template<class K>
+struct hasher : public std::unary_function<K, hash_t>
+{
+  hash_t operator () (const K &k) const { return hash (k); }
+};
+
 #include <lib/hashmap.h>
+#include <lib/hashset.h>
+
 #include <lib/ullmanmap.h>
 #include <lib/mapcommon.h>
 #include <lib/unionfind.h>
 #include <lib/priority_queue.h>
 #include <lib/directed_multigraph.h>
+
+// ??? io
+
+class algebra_writer
+{
+public:
+  // modules
+  unsigned io_id_counter;
+  map<unsigned, unsigned> id_io_id;
+  
+  algebra_writer () : io_id_counter(0) { }
+  ~algebra_writer () { }
+};
+
+class algebra_reader
+{
+public:
+  map<unsigned, unsigned> io_id_id;
+  
+};
diff --git a/lib/map.h b/lib/map.h
index 3be8881..859829a 100644
--- a/lib/map.h
+++ b/lib/map.h
@@ -1,5 +1,5 @@
 
-/* wrapper for stl map */
+/* wrapper for std::map */
 
 template<class K, class V>
 class map : public map_wrapper<std::map<K, V>, K, V>
@@ -11,9 +11,20 @@ class map : public map_wrapper<std::map<K, V>, K, V>
   map (unsigned dummy_size) : base(dummy_size) { }
   map (const map &m) : base(m) { }
   map (copy, const map &m) : base(COPY, m) { }
-  map (copy2, const map &m) : base(COPY2, m) { }
+  map (initializer_list<std::pair<const K, V> > il) : base(il) { }
   map (reader &r) : base(r) { }
   ~map () { }
   
   map &operator = (const map &m) { base::operator = (m); return *this; }
+  map &operator = (initializer_list<std::pair<const K, V> > il)
+  {
+    base::operator = (il);
+    return *this;
+  }
 };
+
+template<class K, class V>
+using map_iter = map_wrapper_iter<std::map<K, V>, K, V>;
+
+template<class K, class V>
+using map_const_iter = map_wrapper_const_iter<std::map<K, V>, K, V>;
diff --git a/lib/map_wrapper.h b/lib/map_wrapper.h
index 762bfc7..e263412 100644
--- a/lib/map_wrapper.h
+++ b/lib/map_wrapper.h
@@ -1,5 +1,5 @@
 
-/* wrapper for stl maps */
+/* wrapper for STL maps */
 
 template<class M, class K, class V> class map_wrapper_iter;
 template<class M, class K, class V> class map_wrapper_const_iter;
@@ -7,14 +7,23 @@ template<class M, class K, class V> class map_wrapper_const_iter;
 template<class M, class K, class V>
 class map_wrapper
 {
- private:
   friend class map_wrapper_iter<M, K, V>;
   friend class map_wrapper_const_iter<M, K, V>;
-  
+
+ protected:
   class map_impl : public refcounted
   {
   public:
     M t;
+    
+  public:
+    map_impl () = default;
+    map_impl (const map_impl &) = delete;
+    map_impl (const M &t_) : t(t_) { }
+    map_impl (initializer_list<std::pair<const K, V> > il) : t(il) { }
+    ~map_impl () = default;
+    
+    map_impl &operator = (const map_impl &) = delete;
   };
   
   ptr<map_impl> impl;
@@ -25,14 +34,24 @@ class map_wrapper
   
  public:
   map_wrapper () : impl(new map_impl) { }
-  map_wrapper (unsigned dummy_size) : impl(new map_impl) { }
   map_wrapper (const map_wrapper &m) : impl(m.impl) { }
-  map_wrapper (copy, const map_wrapper &m) : impl(new map_impl) { impl->t = M (m.impl->t); }
-  map_wrapper (copy2, const map_wrapper &m);
+  map_wrapper (copy, const map_wrapper &m) : impl(new map_impl (m.impl->t)) { }
+  map_wrapper (initializer_list<std::pair<const K, V> > il)
+    : impl(new map_impl (il))
+  { }
   map_wrapper (reader &r);
   ~map_wrapper () { }
   
   map_wrapper &operator = (const map_wrapper &m) { impl = m.impl; return *this; }
+  map_wrapper &operator = (initializer_list<std::pair<const K, V> > il)
+  {
+    impl->t = il;
+    return *this;
+  }
+  
+  // range-based for
+  typename M::const_iterator begin () const { return impl->t.begin (); }
+  typename M::const_iterator end () const { return impl->t.end (); }
   
   /* returns the pair associated to the smallest key */
   pair<K, V> head () const
@@ -130,20 +149,12 @@ class map_wrapper
   }
   
   bool operator == (const map_wrapper &m) const;
+  bool operator != (const map_wrapper &m) const { return !operator == (m); }
   bool operator < (const map_wrapper &m) const;
   
   void write_self (writer &w) const;
 };
 
-template<class M, class K, class V>
-map_wrapper<M, K, V>::map_wrapper (copy2, const map_wrapper &m)
-  : impl(new map_impl)
-{
-  /* Keys are immutable.  Just copy the values. */
-  for (const_iter i = m; i; i ++)
-    push (i.key (), V (COPY, i.val ()));
-}
-
 template<class M, class K, class V>
 map_wrapper<M, K, V>::map_wrapper (reader &r)
   : impl(new map_impl)
@@ -222,19 +233,19 @@ template<class M, class K, class V>
 class map_wrapper_iter
 {
  private:
-  map_wrapper<M, K, V> &m;
+  ptr<typename map_wrapper<M, K, V>::map_impl> impl;
   typename M::iterator i, end;
   bool deleted;
   
  public:
-  map_wrapper_iter (map_wrapper<M, K, V> &m_) : m(m_), i(m_.impl->t.begin ()), end(m_.impl->t.end ()), deleted(0) { }
+  map_wrapper_iter (map_wrapper<M, K, V> &m) : impl(m.impl), i(m.impl->t.begin ()), end(m.impl->t.end ()), deleted(0) { }
   ~map_wrapper_iter () { }
   
   void del ()
   {
     assert (!deleted);
     typename M::iterator iprev = i ++;
-    m.impl->t.erase (iprev);
+    impl->t.erase (iprev);
     deleted = 1;
   }
   
@@ -249,10 +260,11 @@ template<class M, class K, class V>
 class map_wrapper_const_iter
 {
  private:
+  ptr<const typename map_wrapper<M, K, V>::map_impl> impl;
   typename M::const_iterator i, end;
   
  public:
-  map_wrapper_const_iter (const map_wrapper<M, K, V> &m) : i(m.impl->t.begin ()), end(m.impl->t.end ()) { }
+  map_wrapper_const_iter (const map_wrapper<M, K, V> &m) : impl(m.impl), i(m.impl->t.begin ()), end(m.impl->t.end ()) { }
   ~map_wrapper_const_iter () { }
   
   const K &key () const { return i->first; }
diff --git a/lib/pair.h b/lib/pair.h
index fb83a2f..46d8f51 100644
--- a/lib/pair.h
+++ b/lib/pair.h
@@ -17,6 +17,7 @@ class pair
   pair &operator = (const pair &p) { first = p.first; second = p.second; return *this; }
   
   bool operator == (const pair &p) const { return first == p.first && second == p.second; }
+  bool operator != (const pair &p) const { return !operator == (p); }
   bool operator < (const pair &p) const { return (first < p.first
 						  || (first == p.first && second < p.second)); }
   
@@ -53,8 +54,9 @@ public:
   triple (const F &first_, const S &second_, const T &third_)
     : first(first_), second(second_), third(third_)
   { }
+  triple (reader &r);
   ~triple () { }
-
+  
   triple &operator = (const triple &t)
   {
     first = t.first;
@@ -67,6 +69,8 @@ public:
   {
     return first == t.first && second == t.second && third == t.third;
   }
+  bool operator != (const triple &t) const { return !operator == (t); }
+  
   bool operator < (const triple &t) const
   {
     if (first < t.first)
@@ -78,13 +82,11 @@ public:
       return 1;
     else if (second > t.second)
       return 0;
-
-    if (third < t.third)
-      return 1;
-    else if (third > t.third)
-      return 0;
+    
+    return third < t.third;
   }
   
+  void write_self (writer &w) const;
   hash_t hash_self () const
   {
     hash_t h = hash (first);
@@ -92,3 +94,19 @@ public:
     return hash_combine (h, hash (third));
   }
 };
+
+template <class F, class S, class T>
+triple<F, S, T>::triple (reader &r)
+{
+  read (r, first);
+  read (r, second);
+  read (r, third);
+}
+
+template<class F, class S, class T> void
+triple<F, S, T>::write_self (writer &w) const
+{
+  write (w, first);
+  write (w, second);
+  write (w, third);
+}
diff --git a/lib/priority_queue.h b/lib/priority_queue.h
index 6b008f7..c332977 100644
--- a/lib/priority_queue.h
+++ b/lib/priority_queue.h
@@ -18,6 +18,8 @@ class priority_queue
     weighted &operator = (const weighted &w) { v = w.v; pri = w.pri; }
     
     bool operator == (const weighted &w) const { return v == w.v && pri == w.pri; }
+    bool operator != (const weighted &w) const { return !operator == (w); }
+    
     bool operator < (const weighted &w) const { return pri > w.pri || (pri == w.pri && v < w.v); }
   };
   
diff --git a/lib/refcount.h b/lib/refcount.h
index f638e63..b51da0d 100644
--- a/lib/refcount.h
+++ b/lib/refcount.h
@@ -42,10 +42,10 @@ class refcounted
   
  public:
   refcounted () : refcount(0) { }
-  refcounted (const refcounted &rc); // doesn't exist
+  refcounted (const refcounted &rc) = delete;
   ~refcounted () { }
   
-  refcounted &operator = (const refcounted &rc); // doesn't exist
+  refcounted &operator = (const refcounted &rc) = delete;
 };
 
 template<class T>
diff --git a/lib/set.h b/lib/set.h
index 2137655..0f3e08f 100644
--- a/lib/set.h
+++ b/lib/set.h
@@ -1,149 +1,46 @@
 
-/* wrap for std::set */
-
-template<class T> class set_iter;
-template<class T> class set_const_iter;
+/* wrapper for std::set */
 
 template<class T>
-class set
+class set : public set_wrapper<std::set<T>, T>
 {
- private:
-  friend class set_iter<T>;
-  friend class set_const_iter<T>;
-  
-  class set_impl : public refcounted
-  {
-  public:
-    std::set<T> t;
-  };
-  
-  ptr<set_impl> impl;
+  typedef set_wrapper<std::set<T>, T> base;
   
  public:
-  typedef set_iter<T> iter;
-  typedef set_const_iter<T> const_iter;
-  
- public:
-  set () : impl(new set_impl) { }
-  set (unsigned dummy_size) : impl(new set_impl) { }
-  set (const set &s) : impl(s.impl) { }
-  set (copy, const set &s) : impl(new set_impl) { impl->t = std::set<T> (s.impl->t); }
-  set (reader &r);
+  set () { }
+  set (const set &m) : base(m) { }
+  set (copy, const set &m) : base(COPY, m) { }
+  set (initializer_list<T> il) : base(il) { }
+  set (reader &r) : base(r) { }
   ~set () { }
   
-  set &operator = (const set &s) { impl = s.impl; return *this; }
-  
-  T pop ()
-  {
-    typename std::set<T>::const_iterator i = impl->t.begin ();
-    assert (i != impl->t.end ());
-    T tmp = *i;
-    impl->t.erase (i);
-    return tmp;
-  }
-  
-  bool is_empty () const { return impl->t.empty (); }
-  unsigned card () const { return impl->t.size (); }
-  const T &head () const;
-  const T &tail () const;
+  set &operator = (const set &m) { base::operator = (m); return *this; }
+  set &operator = (initializer_list<T> il) { base::operator = (il); }
   
   bool operator == (const set &s) const;
+  bool operator != (const set &s) const { return !operator == (s); }
   bool operator < (const set &s) const;
   bool operator <= (const set &s) const;
   
-  void clear () { impl->t.clear (); }
-  void push (const T &v) { assert (impl->t.find (v) == impl->t.end ()); impl->t.insert (v); }
   bool toggle (const T &v);
-  void operator += (const T &v) { impl->t.insert (v); }
-  void operator -= (const T &v) { impl->t.erase (v); }
-  void yank (const T &v) { assert (operator % (v)); impl->t.erase (v); }
   
-  set &operator |= (const set<T> &s);
-  set &operator &= (const set<T> &s);
-  set &operator ^= (const set<T> &s);
-  
-  bool operator % (const T &v) const { return impl->t.find (v) != impl->t.end (); }
-  bool operator () (const T &v) const { return operator % (v); }
-  
-  void write_self (writer &w) const;
-  hash_t hash_self () const;
+  set &operator |= (const set &s);
+  set &operator &= (const set &s);
+  set &operator ^= (const set &s);
 };
 
 template<class T>
-class set_iter
-{
-  set<T> &s;
-  typename std::set<T>::const_iterator i, end;
-  bool deleted;
-  
- public:
-  set_iter (set<T> &s_) : s(s_), i(s_.impl->t.begin ()), end(s_.impl->t.end ()), deleted(0) { }
-  ~set_iter () { }
-  
-  void del ()
-  {
-    assert (!deleted);
-    typename std::set<T>::const_iterator iprev = i ++;
-    s.impl->t.erase (iprev);
-    deleted = 1;
-  }
-  
-  const T &val () const { assert (!deleted); return *i; }
-  operator bool () const { assert (!deleted); return i != end; }
-  set_iter &operator ++ () { if (deleted) deleted = 0; else i ++; return *this; }
-  void operator ++ (int) { operator ++ (); }
-};
+  using set_iter = set_wrapper_iter<std::set<T>, T>;
 
 template<class T>
-class set_const_iter
-{
-  typename std::set<T>::const_iterator i, end;
-  
- public:
-  set_const_iter (const set<T> &s) : i(s.impl->t.begin ()), end(s.impl->t.end ()) { }
-  ~set_const_iter () { }
-  
-  const T &val () const { return *i; }
-  operator bool () const { return i != end; }
-  set_const_iter &operator ++ () { i ++; return *this; }
-  void operator ++ (int) { i ++; }
-};
-
-template<class T>
-set<T>::set (reader &r)
-{
-  unsigned n;
-  read (r, n);
-  T x;
-  for (unsigned i = 0; i < n; i ++)
-    {
-      read (r, x);
-      push (x);
-    }
-}
-
-template<class T> const T &
-set<T>::head () const
-{
-  set_const_iter<T> i = *this;
-  assert (i);
-  return i.val ();
-}
-
-template<class T> const T &
-set<T>::tail () const
-{
-  typename std::set<T>::const_reverse_iterator i = impl->t.rbegin ();
-  assert (i != impl->t.rend ());
-  return *i;
-}  
+  using set_const_iter = set_wrapper_const_iter<std::set<T>, T>;
 
 template<class T> bool
 set<T>::operator == (const set &s) const
 {
-  typename std::set<T>::const_iterator i = impl->t.begin (),
+  typename std::set<T>::const_iterator i = this->impl->t.begin (),
     j = s.impl->t.begin (),
-    iend = impl->t.end (),
+    iend = this->impl->t.end (),
     jend = s.impl->t.end ();
   
   while (i != iend && j != jend && *i == *j)
@@ -155,9 +52,9 @@ set<T>::operator == (const set &s) const
 template<class T> bool
 set<T>::operator < (const set &s) const
 {
-  typename std::set<T>::const_iterator i = impl->t.begin (),
+  typename std::set<T>::const_iterator i = this->impl->t.begin (),
     j = s.impl->t.begin (),
-    iend = impl->t.end (),
+    iend = this->impl->t.end (),
     jend = s.impl->t.end ();
   
   while (i != iend && j != jend && *i == *j)
@@ -169,9 +66,9 @@ set<T>::operator < (const set &s) const
 template<class T> bool
 set<T>::operator <= (const set &s) const
 {
-  typename std::set<T>::const_iterator i = impl->t.begin (),
+  typename std::set<T>::const_iterator i = this->impl->t.begin (),
     j = s.impl->t.begin (),
-    iend = impl->t.end (),
+    iend = this->impl->t.end (),
     jend = s.impl->t.end ();
   
   while (i != iend && j != jend && *i == *j)
@@ -184,56 +81,56 @@ set<T>::operator <= (const set &s) const
 template<class T> bool
 set<T>::toggle (const T &v)
 {
-  typename std::set<T>::const_iterator i = impl->t.lower_bound (v),
-    end = impl->t.end ();
+  typename std::set<T>::const_iterator i = this->impl->t.lower_bound (v),
+    end = this->impl->t.end ();
   if (i != end && *i == v)
     {
-      impl->t.erase (i);
+      this->impl->t.erase (i);
       return 1;
     }
   else
     {
-      if (i == impl->t.begin ())
-	impl->t.insert (v);
+      if (i == this->impl->t.begin ())
+	this->impl->t.insert (v);
       else
 	{
 	  typename std::set<T>::const_iterator j = --i;
 	  assert (*j < v);
-	  impl->t.insert (j, v);
+	  this->impl->t.insert (j, v);
 	}
       return 0;
     }
 }
 
 template<class T> set<T> &
-set<T>::operator |= (const set<T> &s)
+set<T>::operator |= (const set &s)
 {
   std::set<T> news;
-  std::set_union (impl->t.begin (), impl->t.end (),
+  std::set_union (this->impl->t.begin (), this->impl->t.end (),
 		  s.impl->t.begin (), s.impl->t.end (),
 		  inserter (news, news.begin ()));
-  impl->t = news;
+  this->impl->t = news;
   return *this;
 }
 
 template<class T> set<T> &
-set<T>::operator &= (const set<T> &s)
+set<T>::operator &= (const set &s)
 {
   std::set<T> news;
-  std::set_intersection (impl->t.begin (), impl->t.end (),
+  std::set_intersection (this->impl->t.begin (), this->impl->t.end (),
 			 s.impl->t.begin (), s.impl->t.end (),
 			 inserter (news, news.begin ()));
-  impl->t = news;
+  this->impl->t = news;
   return *this;
 }
 
 template<class T> set<T> &
-set<T>::operator ^= (const set<T> &s)
+set<T>::operator ^= (const set &s)
 {
 #if 0
   printf ("before:\n");
   printf ("this:");
-  for (typename std::set<T>::const_iterator i = impl->t.begin (); i != impl->t.end (); i ++)
+  for (typename std::set<T>::const_iterator i = this->impl->t.begin (); i != this->impl->t.end (); i ++)
     printf (" %d", *i);
   printf ("\n");
   printf ("s:");
@@ -243,25 +140,25 @@ set<T>::operator ^= (const set<T> &s)
 #endif
   
 #if 1
-  typename std::set<T>::const_iterator i = impl->t.begin (),
-    iend = impl->t.end (),
+  typename std::set<T>::const_iterator i = this->impl->t.begin (),
+    iend = this->impl->t.end (),
     j = s.impl->t.begin (),
     jend = s.impl->t.end ();
   
   while (i != iend && j != jend && *i == *j)
     {
-      impl->t.erase (i);
-      i = impl->t.begin ();
+      this->impl->t.erase (i);
+      i = this->impl->t.begin ();
       j ++;
     }
   
   if (i == iend)
-    impl->t.insert (j, jend);
+    this->impl->t.insert (j, jend);
   else
     {
       if (j != jend && *j < *i)
 	{
-	  i = impl->t.insert (*j).first;
+	  i = this->impl->t.insert (*j).first;
 	  j ++;
 	}
       
@@ -274,14 +171,14 @@ set<T>::operator ^= (const set<T> &s)
 	    {
 	      while (j != jend)
 		{
-		  iprev = impl->t.insert (iprev, *j);
+		  iprev = this->impl->t.insert (iprev, *j);
 		  j ++;
 		}
 	      break;
 	    }
 	  else if (*i == *j)
 	    {
-	      impl->t.erase (i);
+	      this->impl->t.erase (i);
 	      i = iprev;
 	      j ++;
 	    }
@@ -289,12 +186,12 @@ set<T>::operator ^= (const set<T> &s)
 	    {
 	      while (j != jend && *j < *i)
 		{
-		  iprev = impl->t.insert (iprev, *j);
+		  iprev = this->impl->t.insert (iprev, *j);
 		  j ++;
 		}
 	      if (j != jend && *j == *i)
 		{
-		  impl->t.erase (i);
+		  this->impl->t.erase (i);
 		  i = iprev;
 		  j ++;
 		}
@@ -305,37 +202,19 @@ set<T>::operator ^= (const set<T> &s)
   
 #if 0
   std::set<T> news;
-  std::set_symmetric_difference (impl->t.begin (), impl->t.end (),
+  std::set_symmetric_difference (this->impl->t.begin (), this->impl->t.end (),
 				 s.impl->t.begin (), s.impl->t.end (),
 				 inserter (news, news.begin ()));
-  impl->t = news;
+  this->impl->t = news;
 #endif
   
 #if 0
   printf ("after:\n");
   printf ("this:");
-  for (typename std::set<T>::const_iterator i = impl->t.begin (); i != impl->t.end (); i ++)
+  for (typename std::set<T>::const_iterator i = this->impl->t.begin (); i != this->impl->t.end (); i ++)
     printf (" %d", *i);
   printf ("\n");
 #endif
   
   return *this;
 }
-
-template<class T> void
-set<T>::write_self (writer &w) const
-{
-  write (w, card ());
-  for (const_iter i = *this; i; i ++)
-    write (w, i.val ());
-}
-
-template<class T> hash_t
-set<T>::hash_self () const
-{
-  hash_t h = hash (card ());
-  unsigned j = 0;
-  for (const_iter i = *this; i && j < 10; i ++, j ++)
-    h = hash_combine (h, hash (i.val ()));
-  return h;
-}
diff --git a/lib/set_wrapper.h b/lib/set_wrapper.h
new file mode 100644
index 0000000..ef8eb04
--- /dev/null
+++ b/lib/set_wrapper.h
@@ -0,0 +1,171 @@
+
+/* wrapper for STL maps */
+
+template<class S, class T> class set_wrapper_iter;
+template<class S, class T> class set_wrapper_const_iter;
+
+template<class S, class T>
+class set_wrapper
+{
+  friend class set_wrapper_iter<S, T>;
+  friend class set_wrapper_const_iter<S, T>;
+  
+ protected:
+  class set_wrapper_impl : public refcounted
+  {
+  public:
+    S t;
+    
+  public:
+    set_wrapper_impl () = default;
+    set_wrapper_impl (const set_wrapper_impl &) = delete;
+    set_wrapper_impl (const S &t_) : t(t_) { }
+    set_wrapper_impl (initializer_list<T> il) : t(il) { }
+    ~set_wrapper_impl () = default;
+    
+    set_wrapper_impl &operator = (const set_wrapper_impl &) = delete;
+  };
+  
+  ptr<set_wrapper_impl> impl;
+  
+ public:
+  typedef set_wrapper_iter<S, T> iter;
+  typedef set_wrapper_const_iter<S, T> const_iter;
+  
+ public:
+  set_wrapper () : impl(new set_wrapper_impl) { }
+  set_wrapper (const set_wrapper &s) : impl(s.impl) { }
+  set_wrapper (copy, const set_wrapper &s)
+    : impl(new set_wrapper_impl (s.impl->t))
+  { }
+  set_wrapper (initializer_list<T> il)
+    : impl(new set_wrapper_impl (il))
+  { }
+  set_wrapper (reader &r);
+  ~set_wrapper () { }
+  
+  set_wrapper &operator = (const set_wrapper &s) { impl = s.impl; return *this; }
+  set_wrapper &operator = (initializer_list<T> li)
+  {
+    impl->t = li;
+  }
+  
+  // range-based for
+  typename S::const_iterator begin () const { return impl->t.begin (); }
+  typename S::const_iterator end () const { return impl->t.end (); }
+  
+  T pop ()
+  {
+    typename S::const_iterator i = impl->t.begin ();
+    assert (i != impl->t.end ());
+    T tmp = *i;
+    impl->t.erase (i);
+    return tmp;
+  }
+  
+  bool is_empty () const { return impl->t.empty (); }
+  unsigned card () const { return impl->t.size (); }
+  const T &head () const;
+  const T &tail () const;
+  
+  void clear () { impl->t.clear (); }
+  void push (const T &v) { assert (impl->t.find (v) == impl->t.end ()); impl->t.insert (v); }
+  void operator += (const T &v) { impl->t.insert (v); }
+  void operator -= (const T &v) { impl->t.erase (v); }
+  void yank (const T &v) { assert (operator % (v)); impl->t.erase (v); }
+  
+  bool operator % (const T &v) const { return impl->t.find (v) != impl->t.end (); }
+  bool operator () (const T &v) const { return operator % (v); }
+  
+  void write_self (writer &w) const;
+  hash_t hash_self () const;
+};
+
+template<class S, class T>
+class set_wrapper_iter
+{
+  ptr<typename set_wrapper<S, T>::set_wrapper_impl> impl;
+  typename S::const_iterator i, end;
+  bool deleted;
+  
+ public:
+  set_wrapper_iter (set_wrapper<S, T> &s) : impl(s.impl), i(s.impl->t.begin ()), end(s.impl->t.end ()), deleted(0) { }
+  ~set_wrapper_iter () { }
+  
+  void del ()
+  {
+    assert (!deleted);
+    typename S::const_iterator iprev = i ++;
+    impl->t.erase (iprev);
+    deleted = 1;
+  }
+  
+  const T &val () const { assert (!deleted); return *i; }
+  operator bool () const { assert (!deleted); return i != end; }
+  set_wrapper_iter &operator ++ () { if (deleted) deleted = 0; else i ++; return *this; }
+  void operator ++ (int) { operator ++ (); }
+};
+
+template<class S, class T>
+class set_wrapper_const_iter
+{
+  ptr<const typename set_wrapper<S, T>::set_wrapper_impl> impl;
+  typename S::const_iterator i, end;
+  
+ public:
+  set_wrapper_const_iter (const set_wrapper<S, T> &s) : impl(s.impl), i(s.impl->t.begin ()), end(s.impl->t.end ()) { }
+  ~set_wrapper_const_iter () { }
+  
+  const T &val () const { return *i; }
+  operator bool () const { return i != end; }
+  set_wrapper_const_iter &operator ++ () { i ++; return *this; }
+  void operator ++ (int) { i ++; }
+};
+
+template<class S, class T>
+set_wrapper<S, T>::set_wrapper (reader &r)
+  : impl(new set_wrapper_impl)
+{
+  unsigned n;
+  read (r, n);
+  T x;
+  for (unsigned i = 0; i < n; i ++)
+    {
+      read (r, x);
+      push (x);
+    }
+}
+
+template<class S, class T> const T &
+set_wrapper<S, T>::head () const
+{
+  set_wrapper_const_iter<S, T> i = *this;
+  assert (i);
+  return i.val ();
+}
+
+template<class S, class T> const T &
+set_wrapper<S, T>::tail () const
+{
+  typename S::const_reverse_iterator i = impl->t.rbegin ();
+  assert (i != impl->t.rend ());
+  return *i;
+}  
+
+template<class S, class T> void
+set_wrapper<S, T>::write_self (writer &w) const
+{
+  write (w, card ());
+  for (const_iter i = *this; i; i ++)
+    write (w, i.val ());
+}
+
+template<class S, class T> hash_t
+set_wrapper<S, T>::hash_self () const
+{
+  hash_t h = hash (card ());
+  unsigned j = 0;
+  for (const_iter i = *this; i && j < 10; i ++, j ++)
+    h = hash_combine (h, hash (i.val ()));
+  return h;
+}
diff --git a/lib/ullmanset.h b/lib/ullmanset.h
index 658fc33..d6599ad 100644
--- a/lib/ullmanset.h
+++ b/lib/ullmanset.h
@@ -2,12 +2,15 @@
 template<unsigned B> class ullmanset_iter;
 template<unsigned B> class ullmanset_const_iter;
 
+template<unsigned B> class ullmanset_const_stl_iter;
+
 template<unsigned B>
 class ullmanset
 {
  private:
   friend class ullmanset_iter<B>;
   friend class ullmanset_const_iter<B>;
+  friend class ullmanset_const_stl_iter<B>;
   
   class keypos
   {
@@ -16,57 +19,56 @@ class ullmanset
     unsigned pos;
   };
   
-  class data
+  class data : public refcounted
   {
   public:
-    unsigned refcount;
     unsigned size;
     unsigned n;
     keypos kp[1];
+    
+  public:
+    inline void delete_at_pos (unsigned p);
   };
   
-  data *d;
-  
-  void ref (data *newd) { assert (!d); d = newd; if (d) d->refcount ++; }
-  void unref ();
-  
-  inline void delete_at_pos (unsigned p);
+  ptr<data> d;
   
  public:
   typedef ullmanset_iter<B> iter;
   typedef ullmanset_const_iter<B> const_iter;
   
  public:
-  ullmanset () : d(0) { }
   ullmanset (unsigned size);
-  ullmanset (const ullmanset &s) : d(0) { ref (s.d); }
+  ullmanset () : ullmanset(0) { }
+  ullmanset (const ullmanset &s) : d(s.d) { }
   ullmanset (copy, const ullmanset &s);
-  // ullmanset (const unsignedset &t);
   ullmanset (const bitset &t);
+  ullmanset (unsigned size, initializer_list<unsigned> il);
   ullmanset (reader &r);
-  ~ullmanset () { unref (); }
+  ~ullmanset () { }
   
-  ullmanset &operator = (const ullmanset &s) { unref (); ref (s.d); return *this; }
-  // ullmanset &operator = (const unsignedset &t);
+  ullmanset &operator = (const ullmanset &s) { d = s.d; return *this; }
   ullmanset &operator = (const bitset &t);
   
-  unsigned size () const { return d ? d->size : 0; }
+  // range-based for
+  inline ullmanset_const_stl_iter<B> begin () const;
+  inline ullmanset_const_stl_iter<B> end () const;
   
-  bool is_empty () const { assert (d); return d->n == 0; }
-  unsigned card () const { assert (d); return d->n; }
-  unsigned head () const { assert (d); assert (d->n > 0); return d->kp[0].key; }
+  unsigned size () const { return d->size; }
+  
+  bool is_empty () const { return d->n == 0; }
+  unsigned card () const { return d->n; }
+  unsigned head () const { assert (d->n > 0); return d->kp[0].key; }
   
   bool operator == (const ullmanset &s);
   bool operator != (const ullmanset &s) { return !operator == (s); }
   
   void restore (unsigned old_card)
   {
-    assert (d);
     assert (d->n >= old_card);
     d->n = old_card;
   }
   
-  void clear () { if (d) { d->n = 0; } }
+  void clear () { d->n = 0; }
   
   inline void push (unsigned k);
   inline void operator += (unsigned k); // ??? should these be inline?
@@ -88,89 +90,75 @@ class ullmanset
   
   bool operator % (unsigned k) const
   {
-    assert (d);
     assert ((k - B) >= 0);
     assert ((k - B) < d->size);
     unsigned p = d->kp[k - B].pos;
     return p < d->n && d->kp[p].key == k;
   }
+  
   bool operator () (unsigned k) const { return operator % (k); }
   
   // always 0-based
-  unsigned nth (unsigned p) const { assert (d); assert (p < d->n); return d->kp[p].key; }
+  unsigned nth (unsigned p) const
+  {
+    assert (p < d->n);
+    return d->kp[p].key;
+  }
   
-  unsigned position (unsigned k) const { assert (operator % (k)); return d->kp[k - B].pos; }
+  unsigned position (unsigned k) const
+  {
+    assert (operator % (k));
+    return d->kp[k - B].pos;
+  }
   
   void write_self (writer &w) const;
 };
 
 template<unsigned B> void
-ullmanset<B>::unref ()
+ullmanset<B>::data::delete_at_pos (unsigned p)
 {
-  if (d && --d->refcount == 0)
-    {
-      delete [] (char *)d;
-      d = 0;
-    }
-}
-
-template<unsigned B> void
-ullmanset<B>::delete_at_pos (unsigned p)
-{
-  assert (d);
-  assert (p < d->n);
-  unsigned n = --d->n;
+  assert (p < n);
+  --n;
   if (p != n)
     {
-      unsigned ell = d->kp[n].key;
-      d->kp[ell - B].pos = p;
-      d->kp[p].key = ell;
+      unsigned ell = kp[n].key;
+      kp[ell - B].pos = p;
+      kp[p].key = ell;
     }
 }
 
 template<unsigned B>
 ullmanset<B>::ullmanset (unsigned size)
-  : d(0)
 {
   data *newd = (data *)new char[sizeof (data)
 				+ sizeof (keypos) * size
 				- sizeof (keypos)];
-  newd->refcount = 0;
+  new (newd) data;
   newd->size = size;
   newd->n = 0;
-  ref (newd);
+  d = newd;
+}
+
+template<unsigned B>
+ullmanset<B>::ullmanset (unsigned size, initializer_list<unsigned> il)
+  : ullmanset(size)
+{
+  for (unsigned i : il)
+    operator += (i);
 }
 
 template<unsigned B>
 ullmanset<B>::ullmanset (copy, const ullmanset &s)
-  : d(0)
+  : ullmanset(s.size ())
 {
-  if (s.d)
-    {
-      unsigned bytes = (sizeof (data)
-			+ sizeof (keypos) * s.d->size
-			- sizeof (keypos));
-      data *newd = (data *)new char[bytes];
-      memcpy (newd, s.d, bytes);
-      newd->refcount = 0;
-      ref (newd);
-    }
+  d->n = s.d->n;
+  memcpy (&d->kp[0], &s.d->kp[0], sizeof (keypos) * d->n);
 }
 
 template<unsigned B>
 ullmanset<B>::ullmanset (reader &r)
-  : d(0)
+  : ullmanset(r.read_unsigned ())
 {
-  unsigned size_ = r.read_unsigned ();
-  if (size_ > 0)
-    {
-      data *newd = (data *)new char[sizeof (data) + sizeof (keypos) * (size_ - 1)];
-      newd->refcount = 0;
-      newd->size = size_;
-      newd->n = 0;
-      ref (newd);
-    }
-  
   unsigned card_ = r.read_unsigned ();
   for (unsigned i = 0; i < card_; i ++)
     push (r.read_unsigned ());
@@ -179,8 +167,6 @@ ullmanset<B>::ullmanset (reader &r)
 template<unsigned B> bool
 ullmanset<B>::operator == (const ullmanset &s)
 {
-  assert (d);
-  assert (s.d);
   if (d->n != s.d->n)
     return 0;
   for (iter i = *this; i; i ++)
@@ -194,7 +180,6 @@ ullmanset<B>::operator == (const ullmanset &s)
 template<unsigned B> void
 ullmanset<B>::push (unsigned k)
 {
-  assert (d);
   assert (!operator % (k));
   unsigned p = d->n ++;
   d->kp[p].key = k;
@@ -204,7 +189,6 @@ ullmanset<B>::push (unsigned k)
 template<unsigned B> void
 ullmanset<B>::operator += (unsigned k)
 {
-  assert (d);
   assert ((k - B) >= 0);
   assert ((k - B) < d->size);
   if (!operator % (k))
@@ -218,13 +202,12 @@ ullmanset<B>::operator += (unsigned k)
 template<unsigned B> void
 ullmanset<B>::operator -= (unsigned k)
 {
-  assert (d);
   assert ((k - B) >= 0);
   assert ((k - B) < d->size);
   if (operator % (k))
     {
       unsigned p = d->kp[k - B].pos;
-      delete_at_pos (p);
+      d->delete_at_pos (p);
     }
 }
   
@@ -270,22 +253,49 @@ ullmanset<B>::write_self (writer &w) const
     write (w, i.val ());
 }
 
+template<unsigned B>
+class ullmanset_const_stl_iter
+{
+ private:
+  ptr<typename ullmanset<B>::data> d;
+  unsigned p;
+  
+ public:
+  ullmanset_const_stl_iter (const ullmanset<B> &s, unsigned p_) : d(s.d), p(p_) { }
+  
+  bool operator != (const ullmanset_const_stl_iter &end) const { return p != end.p; }
+  unsigned operator * () const { assert (p < d->n); return d->kp[p].key; }
+  ullmanset_const_stl_iter &operator ++ () { p ++; return *this; }
+};
+
+template<unsigned B> ullmanset_const_stl_iter<B>
+ullmanset<B>::begin () const
+{
+  return ullmanset_const_stl_iter<B> (*this, 0);
+}
+
+template<unsigned B> ullmanset_const_stl_iter<B>
+ullmanset<B>::end () const
+{
+  return ullmanset_const_stl_iter<B> (*this, d->n);
+}
+
 template<unsigned B>
 class ullmanset_iter
 {
  private:
-  ullmanset<B> &s;
+  ptr<typename ullmanset<B>::data> d;
   unsigned i;
   bool deleted;
   
  public:
-  ullmanset_iter (ullmanset<B> &s_) : s(s_), i(0), deleted(0) { assert (s_.d); }
+  ullmanset_iter (ullmanset<B> &s) : d(s.d), i(0), deleted(0) { }
   ~ullmanset_iter () { }
   
-  void del () { assert (!deleted); assert (i < s.d->n); s.delete_at_pos (i); }
-  unsigned val () const { assert (!deleted); assert (i < s.d->n); return s.d->kp[i].key; }
-  unsigned pos () const { assert (!deleted); assert (i < s.d->n); return i; }
-  operator bool () const { assert (!deleted); return i < s.d->n; }
+  void del () { assert (!deleted); assert (i < d->n); d->delete_at_pos (i); }
+  unsigned val () const { assert (!deleted); assert (i < d->n); return d->kp[i].key; }
+  unsigned pos () const { assert (!deleted); assert (i < d->n); return i; }
+  operator bool () const { assert (!deleted); return i < d->n; }
   void operator ++ () { if (deleted) deleted = 0; else i ++; }
   void operator ++ (int) { operator ++ (); }
 };
@@ -294,16 +304,16 @@ template<unsigned B>
 class ullmanset_const_iter
 {
  private:
-  const ullmanset<B> &s;
+  ptr<typename ullmanset<B>::data> d;
   unsigned i;
   
  public:
-  ullmanset_const_iter (const ullmanset<B> &s_) : s(s_), i(0) { assert (s_.d); }
+  ullmanset_const_iter (const ullmanset<B> &s) : d(s.d), i(0) { }
   ~ullmanset_const_iter () { }
   
-  unsigned val () const { assert (i < s.d->n); return s.d->kp[i].key; }
-  unsigned pos () const { assert (i < s.d->n); return i; }
-  operator bool () const { return i < s.d->n; }
+  unsigned val () const { assert (i < d->n); return d->kp[i].key; }
+  unsigned pos () const { assert (i < d->n); return i; }
+  operator bool () const { return i < d->n; }
   void operator ++ () { i ++; }
   void operator ++ (int) { i ++; }
 };
diff --git a/lib/vector.h b/lib/vector.h
index 52b1f15..06f4e52 100644
--- a/lib/vector.h
+++ b/lib/vector.h
@@ -36,8 +36,10 @@ public:
   
   vector &operator = (const vector &v) { unref (); ref (v.d); return *this; }
   
-  bool operator < (const vector &v) const;
   bool operator == (const vector &v) const;
+  bool operator != (const vector &v) const { return !operator == (v); }
+  
+  bool operator < (const vector &v) const;
   
   bool operator % (unsigned i) { return d && i < d->n; }
   
diff --git a/main.cpp b/main.cpp
index 4ed6dc1..669272b 100644
--- a/main.cpp
+++ b/main.cpp
@@ -60,14 +60,14 @@ test_ring (int p)
 	  
 	  if (x != 0 && x | y)
 	    {
-	      R q = y.div (x);
+	      R q = y.divide_exact (x);
 	      assert (y == q * x);
 	    }
 	  
 	  if (x != 0 || y != 0)
 	    {
-	      triple<R, R, R> t = x.extended_gcd (y);
-	      assert (t.first == t.second*x + t.third*y);
+	      tuple<R, R, R> t = x.extended_gcd (y);
+	      assert (get<0> (t) == get<1> (t)*x + get<2> (t)*y);
 	    }
 	  
 	  for (int k = -n; k <= n; k ++)
@@ -124,58 +124,3412 @@ rank_lte (multivariate_laurentpoly<Z> p,
   return 1;
 }
 
-int
-main ()
+triple<multivariate_laurentpoly<Z>,
+       multivariate_laurentpoly<Z>,
+       multivariate_laurentpoly<Z> >
+square (knot_diagram &kd)
+{
+  cube<Z2> c (kd);
+  mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
+  
+  chain_complex_simplifier<Z2> s (c.khC, d, 1);
+  
+  steenrod_square sq (c, d, s);
+  mod_map<Z2> sq1 = sq.sq1 ();
+  // display ("sq1:\n", sq1);
+  
+  mod_map<Z2> sq2 = sq.sq2 ();
+  // display ("sq2:\n", sq2);
+  
+  assert (sq1.compose (sq1) == 0);
+  assert (sq2.compose (sq2) + sq1.compose (sq2).compose (sq1) == 0);
+  
+  multivariate_laurentpoly<Z> P = s.new_C->free_poincare_polynomial ();
+  
+  ptr<const free_submodule<Z2> > sq1_im = sq1.image ();
+  multivariate_laurentpoly<Z> sq1_P = sq1_im->free_poincare_polynomial ();
+  
+  ptr<const free_submodule<Z2> > sq2_im = sq2.image ();
+  multivariate_laurentpoly<Z> sq2_P = sq2_im->free_poincare_polynomial ();
+  
+  return triple<multivariate_laurentpoly<Z>,
+		multivariate_laurentpoly<Z>,
+		multivariate_laurentpoly<Z> > (P, sq1_P, sq2_P);
+}
+
+void
+compute_show_kh_sq (knot_desc desc
+#if 0
+		    ,
+		    multivariate_laurentpoly<Z> orig_P,
+		    multivariate_laurentpoly<Z> orig_sq1_P,
+		    multivariate_laurentpoly<Z> orig_sq2_P
+#endif
+		    )
+{
+  knot_diagram kd = desc.diagram ();
+  
+  printf ("computing %s...\n", kd.name.c_str ());
+  
+  cube<Z2> c (kd);
+  mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
+  
+  chain_complex_simplifier<Z2> s (c.khC, d, 1);
+  
+  steenrod_square sq (c, d, s);
+  mod_map<Z2> sq1 = sq.sq1 ();
+  mod_map<Z2> sq2 = sq.sq2 ();
+  
+  assert (sq1.compose (sq1) == 0);
+  assert (sq2.compose (sq2) + sq1.compose (sq2).compose (sq1) == 0);
+  
+  multivariate_laurentpoly<Z> P = s.new_C->free_poincare_polynomial ();
+  
+  ptr<const free_submodule<Z2> > sq1_im = sq1.image ();
+  multivariate_laurentpoly<Z> sq1_P = sq1_im->free_poincare_polynomial ();
+  
+  ptr<const free_submodule<Z2> > sq2_im = sq2.image ();
+  multivariate_laurentpoly<Z> sq2_P = sq2_im->free_poincare_polynomial ();
+  
+  printf ("  P "); display (P);
+  printf ("  sq1_P "); display (sq1_P);
+  printf ("  sq2_P "); display (sq2_P);
+  
+#if 0
+  assert (P == orig_P);
+  assert (sq1_P == orig_sq1_P);
+  assert (sq2_P == orig_sq2_P);
+#endif
+}
+
+unsigned
+homological_width (ptr<const module<Z2> > H)
+{
+  int maxd = -1000,
+    mind = 1000;
+  set<grading> gs = H->gradings ();
+  for (set_const_iter<grading> gg = gs; gg; gg ++)
+    {
+      grading hq = gg.val ();
+      int d = 2 * hq.h - hq.q;
+      if (d < mind)
+	mind = d;
+      if (d > maxd)
+	maxd = d;
+    }
+  int dwidth = maxd - mind;
+  
+  assert (is_even (dwidth));
+  unsigned hwidth = (dwidth / 2) + 1;
+  return hwidth;
+}
+
+basedvector<unsigned, 1> hwidth_knots;
+
+void
+load (map<knot_desc,
+	  pair<mod_map<Z2>, mod_map<Z2> > > &knot_kh_sq,
+      knot_desc desc)
+{
+  char buf[1000];
+  if (desc.t == knot_desc::TORUS)
+    {
+      sprintf (buf, "knot_kh_sq/T.dat");
+    }
+  else
+    {
+      unsigned j0 = desc.j;
+      
+      switch (desc.t)
+	{
+	case knot_desc::ROLFSEN:
+	  sprintf (buf, "knot_kh_sq/%d_%d.dat", desc.i, j0);
+	  break;
+
+	case knot_desc::HTW:
+	  sprintf (buf, "knot_kh_sq/K%d_%d.dat", desc.i, j0);
+	  break;
+
+	case knot_desc::MT:
+	  sprintf (buf, "knot_kh_sq/L%d_%d.dat", desc.i, j0);
+	  break;
+
+	default: abort ();
+	}
+    }
+  
+  struct stat stat_buf;
+  if (stat (buf, &stat_buf) != 0)
+    {
+      if (errno == ENOENT)
+	return;
+      
+      stderror ("stat: %s", buf);
+      exit (EXIT_FAILURE);
+    }
+  
+  printf ("loading %s...\n", buf);
+  
+  file_reader r (buf, 1);
+  map<knot_desc,
+      pair<mod_map<Z2>, mod_map<Z2> > > m (r);
+  for (map<knot_desc,
+	   pair<mod_map<Z2>, mod_map<Z2> > >::const_iter i = m; i; i ++)
+    {
+      mod_map<Z2> sq1 = i.val ().first;
+      ptr<const module<Z2> > H = sq1.domain ();
+      unsigned hwidth = homological_width (H);
+      // hwidth_knots[hwidth] ++;
+      
+#if 0
+      if (hwidth == 2)
+	continue;
+      if (i.key ().t == knot_desc::MT
+	  && i.key ().diagram ().num_components () == 1)
+	continue;
+#endif
+      
+      knot_kh_sq.push (i.key (), i.val ());
+    }
+  
+  printf ("done.\n");
+}
+
+void
+load (map<knot_desc,
+	  pair<mod_map<Z2>, mod_map<Z2> > > &knot_kh_sq,
+      const char *file)
+{
+  char buf[1000];
+  sprintf (buf, "knot_kh_sq/%s", file);
+  
+  printf ("loading %s...\n", buf);
+  
+  struct stat stat_buf;
+  if (stat (buf, &stat_buf) != 0)
+    {
+      if (errno == ENOENT)
+	{
+	  printf ("%s does not exist.\n", buf);
+	  return;
+	}
+      
+      stderror ("stat: %s", buf);
+      exit (EXIT_FAILURE);
+    }
+  
+  file_reader r (buf, 1);
+  map<knot_desc,
+      pair<mod_map<Z2>, mod_map<Z2> > > m (r);
+  for (map<knot_desc,
+	   pair<mod_map<Z2>, mod_map<Z2> > >::const_iter i = m; i; i ++)
+    {
+#if 0
+      mod_map<Z2> sq1 = i.val ().first;
+      ptr<const module<Z2> > H = sq1.domain ();
+      unsigned hwidth = homological_width (H);
+      // hwidth_knots[hwidth] ++;
+#endif
+      
+#if 0
+      if (hwidth == 2)
+	continue;
+      if (i.key ().t == knot_desc::MT
+	  && i.key ().diagram ().num_components () == 1)
+	continue;
+#endif
+      
+      knot_kh_sq.push (i.key (), i.val ());
+    }
+  
+  printf ("done.\n");
+}
+
+static const int block_size = 100;
+
+void
+sage_show (FILE *fp,
+	   std::string prefix,
+	   map<grading, basedvector<unsigned, 1> > hq_gens,
+	   ptr<const module<Z2> > H)
+{
+  fprintf (fp, "%s_hom={", prefix.c_str ());
+  bool first = 1;
+  for (map<grading, basedvector<unsigned, 1> >::const_iter i = hq_gens; i; i ++)
+    {
+      if (first)
+	first = 0;
+      else
+	fprintf (fp, ", ");
+      fprintf (fp, "(%d, %d): %d", i.key ().h, i.key ().q, i.val ().size ());
+    }      
+  fprintf (fp, "}\n");
+}
+
+void
+sage_show (FILE *fp,
+	   std::string prefix,
+	   map<grading, basedvector<unsigned, 1> > hq_gens,
+	   std::string name,
+	   grading delta,
+	   mod_map<Z2> f)
+{
+  fprintf (fp, "%s_%s={", prefix.c_str (), name.c_str ());
+  
+  bool first = 1;
+  for (map<grading, basedvector<unsigned, 1> >::const_iter i = hq_gens; i; i ++)
+    {
+      grading from_hq = i.key ();
+      basedvector<unsigned, 1> from_gens = i.val ();
+      
+      grading to_hq = from_hq + delta;
+      if (hq_gens % to_hq)
+	{
+	  basedvector<unsigned, 1> to_gens = hq_gens(to_hq);
+	  
+	  if (first)
+	    first = 0;
+	  else
+	    fprintf (fp, ", ");
+	  fprintf (fp, "(%d, %d): [", from_hq.h, from_hq.q);
+	  bool first2 = 1;
+	  for (unsigned j = 1; j <= from_gens.size (); j ++)
+	    {
+	      unsigned gj = from_gens[j];
+	      
+	      if (first2)
+		first2 = 0;
+	      else
+		fprintf (fp, ", ");
+	      fprintf (fp, "(");
+	      for (unsigned k = 1; k <= to_gens.size (); k ++)
+		{
+		  unsigned gk = to_gens[k];
+		  
+		  if (k > 1)
+		    fprintf (fp, ", ");
+		  if (f[gj](gk) == 1)
+		    fprintf (fp, "1");
+		  else
+		    {
+		      assert (f[gj](gk) == 0);
+		      fprintf (fp, "0");
+		    }
+		}
+	      fprintf (fp, ")");
+	    }
+	  fprintf (fp, "]");
+	}
+    }
+  fprintf (fp, "}\n");
+}
+
+void
+sage_show_khsq (FILE *fp,
+		std::string prefix,
+		ptr<const module<Z2> > H,
+		mod_map<Z2> sq1,
+		mod_map<Z2> sq2)
+{
+  unsigned n = H->dim ();
+  
+  map<grading, basedvector<unsigned, 1> > hq_gens;
+  for (unsigned i = 1; i <= H->dim (); i ++)
+    hq_gens[H->generator_grading (i)].append (i);
+  
+  unsigned t = 0;
+  for (map<grading, basedvector<unsigned, 1> >::const_iter i = hq_gens; i; i ++)
+    t += i.val ().size ();
+  assert (t == n);
+  
+  sage_show (fp, prefix, hq_gens, H);
+  sage_show (fp, prefix, hq_gens, "sq1", grading (1, 0), sq1);
+  sage_show (fp, prefix, hq_gens, "sq2", grading (2, 0), sq2);
+  fprintf (fp, "\n");
+}
+
+void
+test_sage_show ()
+{
+  knot_diagram kd (mt_link (11, 0, 449));
+  show (kd); newline ();
+  
+  planar_diagram pd (kd);
+  pd.display_knottheory ();
+  
+  cube<Z2> c (kd);
+  mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
+  
+  chain_complex_simplifier<Z2> s (c.khC, d, 1);
+  
+  steenrod_square sq (c, d, s);
+  mod_map<Z2> sq1 = sq.sq1 ();
+  mod_map<Z2> sq2 = sq.sq2 ();
+  
+  // display ("sq2:\n", sq2);
+  
+  sage_show_khsq (stdout, "L11n449", sq1.domain (), sq1, sq2);
+}
+
+void
+dump_sage ()
+{
+  char buf[1000];
+  
+#if 0
+  for (unsigned i = 10; i >= 1; i --)
+    {
+      if (i <= 10)
+	{
+	  for (unsigned j = 1; j <= rolfsen_crossing_knots (i); j += block_size)
+	    {
+	      load (knot_kh_sq, knot_desc (knot_desc::ROLFSEN, i, j));
+	    }
+	}
+      ...
+    }
+#endif
+  
+  for (unsigned i = 12; i >= 1; i --)
+    {
+      map<knot_desc,
+	  pair<mod_map<Z2>, mod_map<Z2> > > knot_kh_sq;
+      
+#if 0
+      sprintf (buf, "K%d.sage", i);
+      
+      FILE *fp = fopen (buf, "w");
+      if (fp == 0)
+	{
+	  stderror ("fopen: %s", buf);
+	  exit (EXIT_FAILURE);
+	}
+      
+      for (unsigned j = 1; j <= htw_knots (i); j += block_size)
+	{
+	  load (knot_kh_sq, knot_desc (knot_desc::HTW, i, j));
+	}
+      
+      for (map<knot_desc,
+	     pair<mod_map<Z2>, mod_map<Z2> > >::const_iter k = knot_kh_sq; k; k ++)
+	{
+	  sprintf (buf, "K%s", k.key ().name ().c_str ());
+	  sage_show_khsq (fp, buf, k.val ().first.domain (), k.val ().first, k.val ().second);
+	}
+      
+      fclose (fp);
+#endif
+
+#if 1
+      sprintf (buf, "L%d.sage", i);
+      
+      FILE *fp = fopen (buf, "w");
+      if (fp == 0)
+	{
+	  stderror ("fopen: %s", buf);
+	  exit (EXIT_FAILURE);
+	}
+      
+      if (i <= 13)
+	{
+	  for (unsigned j = 1; j <= mt_links (i); j += block_size)
+	    {
+	      load (knot_kh_sq, knot_desc (knot_desc::MT, i, j));
+	    }
+	}
+      
+      if (i == 14)
+	{
+	  for (unsigned j = 1; j <= mt_links (14, 0); j += block_size)
+	    {
+	      load (knot_kh_sq, knot_desc (knot_desc::MT, 14, mt_links (14, 1) + j));
+	    }
+	}
+      
+      for (map<knot_desc,
+	     pair<mod_map<Z2>, mod_map<Z2> > >::const_iter k = knot_kh_sq; k; k ++)
+	{
+	  sprintf (buf, "K%s", k.key ().name ().c_str ());
+	  sage_show_khsq (fp, buf, k.val ().first.domain (), k.val ().first, k.val ().second);
+	}
+      
+      fclose (fp);
+#endif
+    }
+}
+
+void
+convert_knot_sq ()
+{
+  FILE *fp = open_file ("L14.files", "r");
+  char buf[1000];
+  
+  gzfile_writer w ("L14_sq.dat.gz");
+  w.write_unsigned (76516);
+  
+  set<knot_desc> keys;
+  while (fgets (buf, 1000, fp))
+    {
+      char *p = strchr (buf, '\n');
+      if (p)
+	{
+	  assert (p[1] == 0);
+	  *p = 0;
+	}
+      
+      map<knot_desc,
+	  pair<mod_map<Z2>, mod_map<Z2> > > knot_sq;
+      load (knot_sq, buf);
+      
+      for (map<knot_desc,
+	     pair<mod_map<Z2>, mod_map<Z2> > >::const_iter k = knot_sq; k; k ++)
+	{
+	  write (w, k.key ());
+	  write (w, k.val ());
+	}
+    }
+  
+  // write (w, knot_sq);
+}
+
+#if 0
+void
+test_knot_sq ()
+{
+  const char *knot_sq_files[] = {
+    "knot_sq/rolfsen_sq.dat.gz",
+    "knot_sq/Ksmall_sq.dat.gz",
+    "knot_sq/K11_sq.dat.gz",
+    "knot_sq/K12_sq.dat.gz",
+    "knot_sq/Lsmall_sq.dat.gz",
+    "knot_sq/L11_sq.dat.gz",
+    "knot_sq/L12_sq.dat.gz",
+    0,
+  };
+  
+  for (unsigned i = 0; knot_sq_files[i] != 0; i ++)
+    {
+      gzfile_reader r (knot_sq_files[i]);
+      map<knot_desc,
+	  pair<mod_map<Z2>, mod_map<Z2> > > knot_sq (r);
+      printf ("|knot_sq| = %d\n", knot_sq.card ());
+    }
+}
+
+pair<mod_map<Z2>, mod_map<Z2> > 
+compute_kh_sq (const knot_desc &desc)
+{
+  abort ();
+}
+#endif
+
+template<class R> pair<ptr<const module<R> >, mod_map<R> >
+FU_complex (ptr<const module<R> > CF,
+	    mod_map<R> dF,
+	    mod_map<R> dU)
+{
+  unsigned n = CF->dim ();
+  
+  basedvector<grading, 1> hq (n * 2);
+  for (unsigned i = 1; i <= n; i ++)
+    hq[i] = hq[n + i] = CF->generator_grading (i);
+  
+  ptr<const module<R> > CFU = new explicit_module<R> (2 * n,
+							basedvector<R, 1> (),
+							hq);
+  map_builder<R> b (CFU);
+  for (unsigned i = 1; i <= n; i ++)
+    {
+      for (linear_combination_const_iter<R> j = dF[i]; j; j ++)
+	{
+	  b[i].muladd (j.val (), j.key ());
+	  b[n + i].muladd (j.val (), n + j.key ());
+	}
+      for (linear_combination_const_iter<R> j = dU[i]; j; j ++)
+	{
+	  b[i].muladd (j.val (), n + j.key ());
+	}
+    }
+  mod_map<R> dFU (b);
+  // assert (dFU.compose (dFU) == 0);
+  // dFU.check_sseq_graded ();
+  return pair<ptr<const module<R> >, mod_map<R> > (CFU, dFU);
+}
+
+void
+compute_twistedU ()
 {
 #if 0
+  for (unsigned i = 1; i <= 11; i ++)
+    for (unsigned j = 1; j <= htw_knots (i, 0); j ++)
+      {
+	knot_diagram kd (htw_knot (i, 0, j));
+      ...}
+#endif
+  for (unsigned i = 1; i <= 10; i ++)
+    for (unsigned j = 1; j <= mt_links (i, 0); j ++)
+      {
+	knot_diagram kd (mt_link (i, 0, j));
+	
+	kd.marked_edge = 1;
+	show (kd); newline ();
+	
+	typedef spanning_tree_complex<Z2>::R R;
+	
+	spanning_tree_complex<Z2> c (kd);
+	ptr<const module<R> > C = c.C;
+	
+	mod_map<R> d2 = c.twisted_d2 ();
+	assert (d2.compose (d2) == 0);
+	
+	mod_map<R> d2U = c.twisted_d2Un (1);
+	assert (d2.compose (d2U) == d2U.compose (d2));
+	
+	chain_complex_simplifier<R> s (C, d2, 2);
+	assert (s.new_d == 0);
+	printf ("|s.new_C| = %d\n", s.new_C->dim ());
+	
+	pair<ptr<const module<R> >, mod_map<R> > p = FU_complex (C, d2, d2U);
+	ptr<const module<R> > CFU = p.first;
+	mod_map<R> dFU = p.second;
+	
+	chain_complex_simplifier<R> s2 (CFU, dFU, 2);
+	assert (s2.new_d == 0);
+	printf ("|s2.new_C| = %d\n", s2.new_C->dim ());
+	
+	if (s2.new_C->dim () < 2 * s.new_C->dim ())
+	  printf (" > EXAMPLE\n");
+      }
+}
+
+
+void
+test_forgetful_ss ()
+{
+  typedef fraction_field<polynomial<Z2> > R;
+  
+  for (unsigned i = 1; i <= 12; i ++)
+    for (unsigned j = 1; j <= mt_links (i, 0); j ++)
+      {
+	knot_diagram kd (mt_link (i, 0, j));
+	unsigned n = kd.num_components ();
+	if (n < 2)
+	  continue;
+	
+	show (kd); newline ();
+	
+	unionfind<1> u (kd.num_edges ());
+	for (unsigned i = 1; i <= kd.n_crossings; i ++)
+	  {
+	    u.join (kd.ept_edge (kd.crossings[i][1]),
+		    kd.ept_edge (kd.crossings[i][3]));
+	    u.join (kd.ept_edge (kd.crossings[i][2]),
+		    kd.ept_edge (kd.crossings[i][4]));
+	  }
+	
+	printf ("%d components:\n", n);
+	
+	map<unsigned, unsigned> root_comp;
+	unsigned t = 0;
+	for (unsigned i = 1; i <= kd.num_edges (); i ++)
+	  {
+	    if (u.find (i) == i)
+	      {
+		++ t;
+		root_comp.push (i, t);
+	      }
+	  }
+	assert (t == n);
+	
+	printf ("  kd w: %d\n", kd.writhe ());
+	
+	multivariate_laurentpoly<Z> disj_P = 1;
+	for (unsigned k = 1; k <= n; k ++)
+	  {
+	    knot_diagram comp (SUBLINK, smallbitset (n, unsigned_2pow (k - 1)), kd);
+	    
+	    unsigned w = 0;
+	    for (unsigned i = 1; i <= kd.n_crossings; i ++)
+	      {
+		if (root_comp(u.find (kd.ept_edge (kd.crossings[i][1]))) == k
+		    && root_comp(u.find (kd.ept_edge (kd.crossings[i][2]))) == k)
+		  {
+		    if (kd.is_to_ept (kd.crossings[i][1]) == kd.is_to_ept (kd.crossings[i][4]))
+		      w ++;
+		    else
+		      w --;
+		  }
+	      }
+	    
+	    printf ("  % 2d w: %d\n", k, w);
+	    
+	    cube<R> c (comp);
+	    mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
+	  
+	    chain_complex_simplifier<R> s (c.khC, d, 1);
+	    assert (s.new_d == 0);
+	    
+	    multivariate_laurentpoly<Z> P = s.new_C->free_ell_poincare_polynomial ();
+	    printf ("  % 2d P: ", k);
+	    display (P);
+	    
+	    disj_P *= (P
+		       * multivariate_laurentpoly<Z> (1, VARIABLE, 1, w)
+		       );
+	  }
+	
+	cube<R> c (kd);
+	
+#if 0
+	for (unsigned i = 0; i < c.n_resolutions; i ++)
+	  {
+	    smallbitset state (kd.n_crossings, i);
+	    smoothing s (kd, state);
+	    s.show_self (kd, state);
+	    newline ();
+	  }
+#endif
+	
+	mod_map<R> untwisted_d = c.compute_d (1, 0, 0, 0, 0);
+	assert (untwisted_d.compose (untwisted_d) == 0);
+	
+	chain_complex_simplifier<R> s1 (c.khC, untwisted_d, 1);
+	assert (s1.new_d == 0);
+	
+	multivariate_laurentpoly<Z> P1 = s1.new_C->free_ell_poincare_polynomial ();
+	display ("  link P      : ", P1);
+	
+	display ("  disj_P (adj): ", disj_P);
+	
+	mod_map<R> d = untwisted_d;
+	for (unsigned x = 1; x <= kd.n_crossings; x ++)
+	  {
+	    unsigned r1 = u.find (kd.ept_edge (kd.crossings[x][1])),
+	      r2 = u.find (kd.ept_edge (kd.crossings[x][2]));
+	    
+	    d = d + c.compute_dinv (x)*(R (polynomial<Z2> (1, root_comp(r1))) + 1);
+	    d = d + c.compute_dinv (x)*(R (polynomial<Z2> (1, root_comp(r2))) + 1);
+	    
+#if 0					
+	    R hbar (polynomial<Z2> (1, 1));
+	    R hbarp1 = hbar + 1;
+	    
+	    // if (u.find (kd.ept_edge (kd.crossings[x][1])) != u.find (kd.ept_edge (kd.crossings[x][2])))
+	    
+	    if (u.find (kd.ept_edge (kd.crossings[x][1])) != comp1_root)
+	      d = d + c.compute_dinv (x)*hbarp1;
+	    
+	    if (u.find (kd.ept_edge (kd.crossings[x][2])) != comp1_root)
+	      d = d + c.compute_dinv (x)*hbarp1;
+#endif
+	  }
+	
+#if 0
+	mod_map<R> h = d.compose (d);
+	display ("h:\n", h);
+#endif
+	
+	assert (d.compose (d) == 0);
+	
+	// display ("d:\n", d);
+	
+	chain_complex_simplifier<R> s2 (c.khC, d, -1);
+	assert (s2.new_d == 0);
+
+	multivariate_laurentpoly<Z> P2 = (s2.new_C->free_ell_poincare_polynomial ()
+					  * multivariate_laurentpoly<Z> (1, VARIABLE, 1, kd.writhe ())
+					  );
+	display ("  Einf P (adj): ", P2);
+	
+	if (disj_P == P2)
+	  printf (" disj_P == Einf P (adj): YES!\n");
+	else
+	  printf (" disj_P != Einf P (adj): NO :-(!\n");
+	fflush (stdout);
+      }
+}
+
+void
+compute_forgetful_torsion ()
+{
+  typedef polynomial<Zp<2> > R;
+  
+  for (unsigned i = 1; i <= 12; i ++)
+    for (unsigned j = 1; j <= mt_links (i, 0); j ++)
+      {
+	knot_diagram kd (mt_link (i, 0, j));
+	unsigned n = kd.num_components ();
+	if (n != 2)
+	  continue;
+	
+	show (kd); newline ();
+	
+	unionfind<1> u (kd.num_edges ());
+	for (unsigned i = 1; i <= kd.n_crossings; i ++)
+	  {
+	    u.join (kd.ept_edge (kd.crossings[i][1]),
+		    kd.ept_edge (kd.crossings[i][3]));
+	    u.join (kd.ept_edge (kd.crossings[i][2]),
+		    kd.ept_edge (kd.crossings[i][4]));
+	  }
+	
+	map<unsigned, unsigned> root_comp;
+	unsigned t = 0;
+	for (unsigned i = 1; i <= kd.num_edges (); i ++)
+	  {
+	    if (u.find (i) == i)
+	      {
+		++ t;
+		root_comp.push (i, t);
+	      }
+	  }
+	assert (t == n);
+	
+	cube<R> c (kd);
+#if 0
+	for (unsigned i = 0; i < c.n_resolutions; i ++)
+	  {
+	    smallbitset state (kd.n_crossings, i);
+	    smoothing s (kd, state);
+	    s.show_self (kd, state);
+	    newline ();
+	  }
+#endif
+	
+	mod_map<R> untwisted_d = c.compute_d (1, 0, 0, 0, 0);
+	assert (untwisted_d.compose (untwisted_d) == 0);
+	
+	mod_map<R> d = untwisted_d;
+	for (unsigned x = 1; x <= kd.n_crossings; x ++)
+	  {
+	    unsigned r1 = u.find (kd.ept_edge (kd.crossings[x][1])),
+	      r2 = u.find (kd.ept_edge (kd.crossings[x][2]));
+	    
+	    if (r1 != r2)
+	      d = d + c.compute_dinv (x)*(R (1, 1));
+	  }
+	
+#if 0
+	mod_map<R> h = d.compose (d);
+	display ("h:\n", h);
+#endif
+	
+	assert (d.compose (d) == 0);
+	
+	// display ("d:\n", d);
+	
+	chain_complex_simplifier<R> s (c.khC, d, 0);
+	printf ("|s.new_C| = %d\n", s.new_C->dim ());
+	display ("s.new_d:\n", s.new_d);
+	
+#if 0
+	ptr<const free_submodule<R> > ker = s.new_d.kernel ();
+	printf ("|ker| = %d\n", ker->dim ());
+	
+	ptr<const free_submodule<R> > im = s.new_d.image ();
+	printf ("|im| = %d\n", im->dim ());
+#endif
+	
+	ptr<const module<R> > H = s.new_d.homology ();
+	display ("H:\n", *H);
+	// printf ("dim H = %d, free rank H = %d\n", H->dim (), H->free_rank ());
+      }
+}
+
+void
+test_forgetful_signs ()
+{
+  typedef Q R;
+  
+#if 1
+  for (unsigned i = 1; i <= 14; i ++)
+    for (unsigned j = 1; j <= mt_links (i, 1); j ++)
+      {
+	knot_diagram kd (mt_link (i, 1, j));
+#endif
+#if 0
+  for (unsigned i = 2; i <= 16; i ++)
+    for (unsigned j = i; j <= 16; j ++)
+      {
+	knot_diagram kd (torus_knot (i, j));
+	assert (kd.n_crossings == (i - 1) * j
+		&& kd.n_crossings <= i * (j - 1));
+	if (kd.n_crossings > 13)
+	  continue;
+      }
+#endif
+	
+	unsigned n = kd.num_components ();
+	if (n < 2)
+	  continue;
+	assert (n < 97);
+	
+	show (kd); newline ();
+	
+	unionfind<1> u (kd.num_edges ());
+	for (unsigned i = 1; i <= kd.n_crossings; i ++)
+	  {
+	    u.join (kd.ept_edge (kd.crossings[i][1]),
+		    kd.ept_edge (kd.crossings[i][3]));
+	    u.join (kd.ept_edge (kd.crossings[i][2]),
+		    kd.ept_edge (kd.crossings[i][4]));
+	  }
+	
+	map<unsigned, unsigned> root_comp;
+	unsigned t = 0;
+	for (unsigned i = 1; i <= kd.num_edges (); i ++)
+	  {
+	    if (u.find (i) == i)
+	      {
+		++ t;
+		root_comp.push (i, t);
+	      }
+	  }
+	assert (t == n);
+	
+	basedvector<R, 1> comp_weight (n);
+	for (unsigned i = 1; i <= n; i ++)
+	  comp_weight[i] = R ((int)i);
+	
+	map<unsigned, R> crossing_over_sign;
+	
+	// crossings
+	set<unsigned> pending;
+	set<unsigned> finished;
+	
+	crossing_over_sign.push (1, R (1));
+	pending.push (1);
+  
+	while (pending.card () > 0)
+	  {
+	    unsigned x = pending.pop ();
+	    finished.push (x);
+      
+	    R s = crossing_over_sign(x);
+      
+	    for (unsigned j = 1; j <= 4; j ++)
+	      {
+		unsigned p = kd.crossings[x][j];
+		R t = kd.is_over_ept (p) ? s : -s;  // sign of (x, p)
+	  
+		unsigned q = kd.edge_other_ept (p);
+		unsigned x2 = kd.ept_crossing[q];
+	  
+		R u = kd.is_over_ept (q) ? -t : t;
+	  
+		if (crossing_over_sign % x2)
+		  assert (crossing_over_sign(x2) == u);
+		else
+		  crossing_over_sign.push (x2, u);
+	  
+		if (! (finished % x2))
+		  pending += x2;
+	      }
+	  }
+	assert (finished.card () == kd.n_crossings);
+	
+#if 0
+	printf ("crossing_over_sign:\n");
+	for (unsigned i = 1; i <= kd.n_crossings; i ++)
+	  {
+	    printf ("  % 2d: ", i);
+	    display (crossing_over_sign(i));
+	  }
+#endif
+	
+	cube<R> c (kd);
+#if 0
+	for (unsigned i = 0; i < c.n_resolutions; i ++)
+	  {
+	    smallbitset state (kd.n_crossings, i);
+	    smoothing s (kd, state);
+	    s.show_self (kd, state);
+	    newline ();
+	  }
+#endif
+  
+	mod_map<R> untwisted_d = c.compute_d (1, 0, 0, 0, 0);
+	assert (untwisted_d.compose (untwisted_d) == 0);
+	
+	mod_map<R> d = untwisted_d;
+	for (unsigned x = 1; x <= kd.n_crossings; x ++)
+	  {
+	    unsigned p1 = kd.crossings[x][1],
+	      p2 = kd.crossings[x][2];
+	    assert (kd.is_over_ept (p2));
+	    
+	    unsigned r1 = u.find (kd.ept_edge (p1)),
+	      r2 = u.find (kd.ept_edge (p2));
+	    
+	    unsigned c1 = root_comp(r1),
+	      c2 = root_comp(r2);
+	    
+	    if (c1 != c2)
+	      {
+		R s = crossing_over_sign(x);
+		
+		R w_under = comp_weight[c1];
+		R w_over = comp_weight[c2];
+		
+		d = d + c.compute_dinv (x)*(s*(w_over - w_under))
+		  ;
+	      }
+	  }
+	
+	// display ("d:\n", d);
+	
+	// mod_map<Z> h = d.compose (d);
+	// display ("h:\n", h);
+	
+	assert (d.compose (d) == 0);
+	
+	ptr<const module<R> > Ek = c.khC;
+	mod_map<R> dk = d;
+	
+	unsigned kinf;
+	for (int dq = 0;; dq -= 2)
+	  {
+	    chain_complex_simplifier<R> s (Ek, dk, dq);
+	    Ek = s.new_C;
+	    dk = s.new_d;
+           
+	    printf ("|E%d| = %d\n", ((- dq) / 2) + 1, Ek->dim ());
+	    if (dk == 0)
+	      {
+		kinf = ((- dq) / 2);
+		break;
+	      }
+	  }
+	
+	unsigned total_lk = kd.total_linking_number ();
+	
+	unsigned trivial_bound = total_lk == 0 ? 2 : total_lk;
+	
+	printf ("kinf = %d, trivial bound = %d (total_lk = %d)\n",
+		kinf, trivial_bound, total_lk);
+	
+	if (trivial_bound < kinf)
+         printf (" > BETTER\n");
+      }
+}
+
+template<class R> sseq
+compute_forgetfulss (knot_diagram &kd,
+		     basedvector<R, 1> comp_weight)
+{
+  unsigned n = kd.num_components ();
+  
+  unionfind<1> u (kd.num_edges ());
+  for (unsigned i = 1; i <= kd.n_crossings; i ++)
+    {
+      u.join (kd.ept_edge (kd.crossings[i][1]),
+	      kd.ept_edge (kd.crossings[i][3]));
+      u.join (kd.ept_edge (kd.crossings[i][2]),
+	      kd.ept_edge (kd.crossings[i][4]));
+    }
+	
+  map<unsigned, unsigned> root_comp;
+  unsigned t = 0;
+  for (unsigned i = 1; i <= kd.num_edges (); i ++)
+    {
+      if (u.find (i) == i)
+	{
+	  ++ t;
+	  root_comp.push (i, t);
+	}
+    }
+  assert (t == n);
+  
+  assert (comp_weight.size () == n);
+  
+  map<unsigned, R> crossing_over_sign;
+	
+  // crossings
+  set<unsigned> pending;
+  set<unsigned> finished;
+	
+  crossing_over_sign.push (1, R (1));
+  pending.push (1);
+  
+  while (pending.card () > 0)
+    {
+      unsigned x = pending.pop ();
+      finished.push (x);
+      
+      R s = crossing_over_sign(x);
+      
+      for (unsigned j = 1; j <= 4; j ++)
+	{
+	  unsigned p = kd.crossings[x][j];
+	  R t = kd.is_over_ept (p) ? s : -s;  // sign of (x, p)
+	  
+	  unsigned q = kd.edge_other_ept (p);
+	  unsigned x2 = kd.ept_crossing[q];
+	  
+	  R u = kd.is_over_ept (q) ? -t : t;
+	  
+	  if (crossing_over_sign % x2)
+	    assert (crossing_over_sign(x2) == u);
+	  else
+	    crossing_over_sign.push (x2, u);
+	  
+	  if (! (finished % x2))
+	    pending += x2;
+	}
+    }
+  assert (finished.card () == kd.n_crossings);
+  
+  cube<R> c (kd);
+#if 0
+  for (unsigned i = 0; i < c.n_resolutions; i ++)
+    {
+      smallbitset state (kd.n_crossings, i);
+      smoothing s (kd, state);
+      s.show_self (kd, state);
+      newline ();
+    }
+#endif
+  
+  mod_map<R> untwisted_d = c.compute_d (1, 0, 0, 0, 0);
+  assert (untwisted_d.compose (untwisted_d) == 0);
+	
+  mod_map<R> d = untwisted_d;
+  for (unsigned x = 1; x <= kd.n_crossings; x ++)
+    {
+      unsigned p1 = kd.crossings[x][1],
+	p2 = kd.crossings[x][2];
+      assert (kd.is_over_ept (p2));
+	    
+      unsigned r1 = u.find (kd.ept_edge (p1)),
+	r2 = u.find (kd.ept_edge (p2));
+	    
+      unsigned c1 = root_comp(r1),
+	c2 = root_comp(r2);
+	    
+      if (c1 != c2)
+	{
+	  R s = crossing_over_sign(x);
+		
+	  R w_under = comp_weight[c1];
+	  R w_over = comp_weight[c2];
+		
+	  d = d + c.compute_dinv (x)*(s*(w_over - w_under))
+	    ;
+	}
+    }
+  
+  assert (d.compose (d) == 0);
+  
+  ptr<const module<R> > C = c.khC;
+  // d
+  
+  int minh = 1000,
+    maxh = -1000,
+    minq = 1000,
+    maxq = -1000;
+  
+  for (unsigned i = 1; i <= C->dim (); i ++)
+    {
+      grading hq = C->generator_grading (i);
+      if (hq.h < minh)
+	minh = hq.h;
+      if (hq.h > maxh)
+	maxh = hq.h;
+      if (hq.q < minq)
+	minq = hq.q;
+      if (hq.q > maxq)
+	maxq = hq.q;
+    }
+  
+  sseq_bounds bounds (minh, maxh, minq, maxq);
+  
+  basedvector<sseq_page, 1> pages;
+  
+  for (unsigned dq = 0;;)
+    {
+      chain_complex_simplifier<R> s (C, d, dq);
+      C = s.new_C;
+      d = s.new_d;
+      
+      dq -= 2;
+      
+      sseq_page pg (bounds);
+      for (unsigned i = 1; i <= C->dim (); i ++)
+	{
+	  grading hq = C->generator_grading (i);
+	  pg.rank[hq.h - bounds.minh][hq.q - bounds.minq] ++;
+	}
+      pages.append (pg);
+      
+#if 0
+      mod_map<R> dk = d.graded_piece (grading (dq + 1, dq));
+      dk.check_grading (grading (dq + 1, dq));
+      
+      ptr<const free_submodule<R> > dk_im = dk.image ();
+      for (unsigned i = 1; i <= dk_im->dim (); i ++)
+	{
+	  grading hq = dk_im->generator_grading (i);
+	  pg.im_rank[hq.h - bounds.minh][hq.q - bounds.minq] ++;
+	}
+#endif
+      
+#if 0
+      printf ("E_%d: ", (-dq) / 2);
+      display (C->free_poincare_polynomial ());
+#endif
+      
+      if (d == 0)
+	break;
+    }
+
+  abort ();
+  // ???
+  return sseq (bounds, grading (0, 0), grading (0, 0), pages);
+}
+
+void
+compute_lee_bound ()
+{
+  typedef Z2 R;
+  
+#if 0
+  for (unsigned i = 2; i <= 16; i ++)
+    for (unsigned j = i; j <= 16; j ++)
+      {
+	knot_diagram kd (torus_knot (i, j));
+	assert (kd.n_crossings == (i - 1) * j
+		&& kd.n_crossings <= i * (j - 1));
+	if (kd.n_crossings > 13)
+	  continue;
+      }
+#endif
+#if 1
   for (unsigned i = 1; i <= 10; i ++)
     for (unsigned j = 1; j <= rolfsen_crossing_knots (i); j ++)
       {
 	knot_diagram kd (rolfsen_knot (i, j));
 #endif
+	
+	show (kd); newline ();
+	
+	cube<R> c (kd);
+#if 0
+	for (unsigned i = 0; i < c.n_resolutions; i ++)
+	  {
+	    smallbitset state (kd.n_crossings, i);
+	    smoothing s (kd, state);
+	    s.show_self (kd, state);
+	    newline ();
+	  }
+#endif
+	
+	mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
+	assert (d.compose (d) == 0);
+	
+	for (unsigned x = 1; x <= kd.n_crossings; x ++)
+	  d = d + c.H_i (x);
+	
+	assert (d.compose (d) == 0);
+	
+	ptr<const module<R> > Ek = c.khC;
+	mod_map<R> dk = d;
+	
+	unsigned kinf;
+	for (int dq = 0;; dq += 2)
+	  {
+	    chain_complex_simplifier<R> s (Ek, dk, dq);
+	    Ek = s.new_C;
+	    dk = s.new_d;
+	    
+	    printf ("|E%d| = %d\n", (dq / 2) + 1, Ek->dim ());
+	    if (dk == 0)
+	      {
+		kinf = (dq / 2);
+		break;
+	      }
+	  }
+	
+	printf ("kinf = %d\n", kinf);
+      }
+}
+
+void
+test_crossing_ss ()
+{
+  typedef multivariate_laurentpoly<Z2> R;
+  
+  for (unsigned i = 1; i <= 14; i ++)
+    for (unsigned j = 1; j <= htw_knots (i); j ++)
+      {
+	knot_diagram kd (htw_knot (i, j));
+	show (kd); newline ();
+	
+	cube<R> c (kd);
+#if 0
+	for (unsigned i = 0; i < c.n_resolutions; i ++)
+	  {
+	    smallbitset state (kd.n_crossings, i);
+	    smoothing s (kd, state);
+	    s.show_self (kd, state);
+	    newline ();
+	  }
+#endif
+	
+	mod_map<R> untwisted_d = c.compute_d (1, 0, 0, 0, 0);
+	assert (untwisted_d.compose (untwisted_d) == 0);
+	
+	mod_map<R> d = untwisted_d;
+	for (unsigned x = 1; x <= kd.n_crossings; x ++)
+	  {
+	    d = d + c.compute_dinv (x)*(R (1, VARIABLE, x));
+	  }
+	
+	display ("d:\n", d);
+	
+	mod_map<R> h = d.compose (d);
+	display ("h:\n", h);
+      }
+}
+
+/* Kh homotopy type:
+   - check for CP^2,
+   - mutation invariance
+   - K-theory */
+
+map<grading, basedvector<int, 1> >
+compute_st (mod_map<Z2> sq1,
+	    mod_map<Z2> sq2)
+{
+  ptr<const module<Z2> > H = sq1.domain ();
+  
+  map<grading, basedvector<int, 1> > st;
+  
+  bool first = 1;
+  set<grading> gs = H->gradings ();
+  for (set_const_iter<grading> gg = gs; gg; gg ++)
+    {
+      grading hq = gg.val (),
+	h1q (hq.h + 1, hq.q),
+	h2q (hq.h + 2, hq.q);
+      
+      ptr<const free_submodule<Z2> > H_hq = H->graded_piece (hq),
+	H_h1q = H->graded_piece (h1q),
+	H_h2q = H->graded_piece (h2q);
+      
+      mod_map<Z2> S = sq2.restrict (H_hq, H_h2q),
+	A = sq1.restrict (H_hq, H_h1q),
+	B = sq1.restrict (H_h1q, H_h2q);
+      
+      ptr<const free_submodule<Z2> > S_im = S.image (),
+	A_ker = A.kernel (),
+	B_im = B.image ();
+      ptr<const free_submodule<Z2> > inter = S_im->intersection (B_im);
+      
+      mod_map<Z2> S_res = S.restrict_from (A_ker);
+      ptr<const free_submodule<Z2> > S_res_im = S_res.image ();
+      
+      ptr<const free_submodule<Z2> > res_inter = S_res_im->intersection (B_im);
+      
+      int r1 = S_im->dim ();
+      int r2 = S_res_im->dim ();
+      int r3 = inter->dim ();
+      int r4 = res_inter->dim ();
+      
+      int s1 = r2 - r4,
+	s2 = r1 - r2 - r3 + r4,
+	s3 = r4,
+	s4 = r3 - r4;
+      
+      if (!s1 && !s2 && !s3 && !s4)
+	continue;
+      
+      basedvector<int, 1> s (4);
+      s[1] = s1;
+      s[2] = s2;
+      s[3] = s3;
+      s[4] = s4;
+      
+      st.push (hq, s);
+    }
+  
+  return st;
+}
+
+void
+display_st (map<grading, basedvector<int, 1> > st)
+{
+  bool first = 1;
+  for (map_const_iter<grading, basedvector<int, 1> > i = st; i; i ++)
+    {
+      grading hq = i.key ();
+      int s1 = i.val ()[1],
+	s2 = i.val ()[2],
+	s3 = i.val ()[3],
+	s4 = i.val ()[4];
+      if (s1 == 0 && s2 == 0 && s3 == 0 && s4 == 0)
+	continue;
+      if (first)
+	first = 0;
+      else
+	printf (",");
+      printf ("$(%d, %d) \\mapsto (%d, %d, %d, %d)$", hq.h, hq.q, s1, s2, s3, s4);
+    }
+  newline ();
+}
+
+void
+test_kh_htt_stronger ()
+{
+  map<pair<multivariate_laurentpoly<Z>,
+	   multivariate_laurentpoly<Z> >,
+      pair<knot_desc,
+	   map<grading, basedvector<int, 1> > > > P_sq1_knot_st;
+  
+  const char *knot_sq_files[] = {
+    "knot_sq/rolfsen_sq.dat.gz",
+    "knot_sq/Ksmall_sq.dat.gz",
+    "knot_sq/K11_sq.dat.gz",
+    "knot_sq/K12_sq.dat.gz",
+    "knot_sq/K13_sq.dat.gz",
+    "knot_sq/K14_sq.dat.gz",
+    "knot_sq/Lsmall_sq.dat.gz",
+    "knot_sq/L11_sq.dat.gz",
+    "knot_sq/L12_sq.dat.gz",
+    "knot_sq/L13_sq.dat.gz",
+    "knot_sq/L14_sq.dat.gz",
+    0,
+  };
+  
+  for (unsigned i = 0; knot_sq_files[i]; i ++)
+    {
+      gzfile_reader r (knot_sq_files[i]);
+      
+      printf ("loading %s...\n", knot_sq_files[i]);
+      
+      unsigned n = r.read_unsigned ();
+      for (unsigned i = 1; i <= n; i ++)
+	{
+	  knot_desc desc (r);
+	  pair<mod_map<Z2>, mod_map<Z2> > p1 (r);
+	  
+	  if (desc.t == knot_desc::MT
+	      || desc.t == knot_desc::MT_ALT
+	      || desc.t == knot_desc::MT_NONALT)
+	    {
+	      knot_diagram kd = desc.diagram ();
+	      if (kd.num_components () == 1)
+		continue;
+	    }
+	  
+	  mod_map<Z2> sq1 = p1.first,
+	    sq2 = p1.second;
+	  ptr<const module<Z2> > H = sq1.domain ();
+	  multivariate_laurentpoly<Z> P = H->free_poincare_polynomial ();
+	  
+	  unsigned hwidth = homological_width (H);
+	  
+	  map<grading, basedvector<int, 1> > st
+	    = compute_st (sq1, sq2);
+	  
+	  ptr<const free_submodule<Z2> > sq1_im = sq1.image ();
+	  multivariate_laurentpoly<Z> sq1_im_P = sq1_im->free_poincare_polynomial ();
+	  
+	  pair<pair<knot_desc,
+		    map<grading, basedvector<int, 1> > > &,
+	       bool> p2 = P_sq1_knot_st.find (pair<multivariate_laurentpoly<Z>,
+					           multivariate_laurentpoly<Z> > (P, sq1_im_P));
+	  if (p2.second)
+	    {
+	      if (p2.first.second != st)
+		{
+		  printf ("DIFFER:\n");
+		  printf ("hwidth = %d\n", hwidth);
+
+		  show (p2.first.first);
+		  display_st (p2.first.second);
+		  
+		  show (desc);
+		  display_st (st);
+		  
+		  printf ("Kh[");
+		  planar_diagram (p2.first.first.diagram ()).show_knottheory ();
+		  printf (", Modulus -> Null][q,t] === Kh[");
+		  planar_diagram (desc.diagram ()).show_knottheory ();
+		  printf (", Modulus -> Null][q,t]\n");
+		}
+	    }
+	  else
+	    {
+	      p2.first.first = desc;
+	      p2.first.second = st;
+	    }
+	}
+    }
+}
+
+void
+find_width4_in_h0 ()
+{
+  const char *knot_sq_files[] = {
+    "knot_sq/rolfsen_sq.dat.gz",
+    "knot_sq/Ksmall_sq.dat.gz",
+    "knot_sq/K11_sq.dat.gz",
+    "knot_sq/K12_sq.dat.gz",
+    "knot_sq/K13_sq.dat.gz",
+    "knot_sq/K14_sq.dat.gz",
+    "knot_sq/K15_sq_part.dat.gz",
+    0,
+  };
+  
+  for (unsigned i = 0; knot_sq_files[i]; i ++)
+    {
+      gzfile_reader r (knot_sq_files[i]);
+      
+      printf ("loading %s...\n", knot_sq_files[i]);
+      fflush (stdout);
+      
+      unsigned n = r.read_unsigned ();
+      for (unsigned i = 1; i <= n; i ++)
+	{
+	  knot_desc desc (r);
+	  // show (desc); newline ();
+	  
+	  pair<mod_map<Z2>, mod_map<Z2> > p (r);
+	  
+	  mod_map<Z2> sq1 = p.first,
+	    sq2 = p.second;
+	  ptr<const module<Z2> > H = sq1.domain ();
+	  
+	  int minq = 1000,
+	    maxq = -1000;
+	  
+	  for (unsigned j = 1; j <= H->dim (); j ++)
+	    {
+	      grading hq = H->generator_grading (j);
+	      if (hq.h == 0)
+		{
+		  if (hq.q < minq)
+		    minq = hq.q;
+		  if (hq.q > maxq)
+		    maxq = hq.q;
+		}
+	    }
+	  
+	  unsigned h0_width = 0;
+	  if (minq <= maxq)
+	    h0_width = (maxq - minq) / 2 + 1;
+	  
+	  // printf ("h0_width = %d\n", h0_width);
+	  
+	  if (h0_width >= 4)
+	    {
+	      show (desc); newline ();
+	      // printf (" > WIDTH 4!!\n");
+	    }
+	}
+    }
+}
+
+void
+test_knot_sq ()
+{
+  const char *knot_sq_files[] = {
+    "knot_sq/rolfsen_sq.dat.gz",
+    "knot_sq/Ksmall_sq.dat.gz",
+    "knot_sq/K11_sq.dat.gz",
+    "knot_sq/K12_sq.dat.gz",
+    "knot_sq/K13_sq.dat.gz",
+    "knot_sq/K14_sq.dat.gz",
+    "knot_sq/K15_sq_part.dat.gz",
+    "knot_sq/Lsmall_sq.dat.gz",
+    "knot_sq/L11_sq.dat.gz",
+    "knot_sq/L12_sq.dat.gz",
+    "knot_sq/L13_sq.dat.gz",
+    "knot_sq/L14_sq.dat.gz",
+    0,
+  };
+  
+  hashset<knot_desc> mutants;
+  for (unsigned i = 11; i <= 15; i ++)
+    {
+      basedvector<basedvector<unsigned, 1>, 1> groups
+	= mutant_knot_groups (i);
+      for (unsigned j = 1; j <= groups.size (); j ++)
+	{
+	  for (unsigned k = 1; k <= groups[j].size (); k ++)
+	    mutants += knot_desc (knot_desc::HTW, i, groups[j][k]);
+	}
+    }
+  printf ("|mutants| = %d\n", mutants.card ());
+  fflush (stdout);
+  
+  basedvector<unsigned, 1> width_hist (7);
+  for (unsigned i = 1; i <= width_hist.size (); i ++)
+    width_hist[i] = 0;
+  
+  hashmap<knot_desc,
+	  pair<mod_map<Z2>, mod_map<Z2> > > mutant_knot_sq,
+    mutant_mknot_sq;
+
+  {
+    gzfile_reader r ("knot_sq/mut_mknot.dat.gz");
+    mutant_mknot_sq = hashmap<knot_desc,
+			      pair<mod_map<Z2>, mod_map<Z2> > > (r);
+    
+    for (hashmap<knot_desc,
+	   pair<mod_map<Z2>, mod_map<Z2> > >::const_iter i = mutant_mknot_sq; i; i ++)
+      {
+	knot_desc desc = i.key ();
+	
+	// careful: duplicate code!
+	pair<mod_map<Z2>, mod_map<Z2> > p = i.val ();
+	
+	mod_map<Z2> sq1 = p.first,
+	  sq2 = p.second;
+	ptr<const module<Z2> > H = sq1.domain ();
+	
+	unsigned w = homological_width (H);
+	assert (w >= 1 && w <= 7);
+	width_hist[w] ++;
+	
+	assert (sq1.compose (sq1) == 0);
+	assert (sq2.compose (sq2) == sq1.compose (sq2).compose (sq1));
+	
+	map<grading, basedvector<int, 1> > st
+	  = compute_st (sq1, sq2);
+	for (map_const_iter<grading, basedvector<int, 1> > i = st; i; i ++)
+	  {
+	    if (i.val ()[1] != 0)
+	      {
+		show (desc); newline ();
+		printf (" > has CP^2\n");
+		fflush (stdout);
+	      }
+	  }
+      }
+  }
+  
+  for (unsigned i = 0; knot_sq_files[i]; i ++)
+    {
+      gzfile_reader r (knot_sq_files[i]);
+      
+      printf ("loading %s...\n", knot_sq_files[i]);
+      fflush (stdout);
+      
+      unsigned n = r.read_unsigned ();
+      for (unsigned i = 1; i <= n; i ++)
+	{
+	  knot_desc desc (r);
+	  
+	  pair<mod_map<Z2>, mod_map<Z2> > p (r);
+	  if (mutants % desc)
+	    mutant_knot_sq.push (desc, p);
+	  
+	  mod_map<Z2> sq1 = p.first,
+	    sq2 = p.second;
+	  ptr<const module<Z2> > H = sq1.domain ();
+	  
+	  unsigned w = homological_width (H);
+	  assert (w >= 1 && w <= 7);
+	  width_hist[w] ++;
+	  
+	  assert (sq1.compose (sq1) == 0);
+	  assert (sq2.compose (sq2) == sq1.compose (sq2).compose (sq1));
+	  
+	  map<grading, basedvector<int, 1> > st
+	    = compute_st (sq1, sq2);
+	  for (map_const_iter<grading, basedvector<int, 1> > i = st; i; i ++)
+	    {
+	      if (i.val ()[1] != 0)
+		{
+		  show (desc); newline ();
+		  printf (" > has CP^2\n");
+		  fflush (stdout);
+		}
+	    }
+	}
+    }
+  
+  printf ("|mutant_knot_sq| = %d\n", mutant_knot_sq.card ());
+  printf ("width_hist:\n");
+  for (unsigned i = 1; i <= width_hist.size (); i ++)
+    {
+      printf (" % 2d: %d\n", i, width_hist[i]);
+    }
+  fflush (stdout);
+  
+  unsigned missing = 0,
+    compared = 0;
+  for (unsigned i = 11; i <= 15; i ++)
+    {
+      basedvector<basedvector<unsigned, 1>, 1> groups
+	= mutant_knot_groups (i);
+      
+      for (unsigned j = 1; j <= groups.size (); j ++)
+	{
+	  bool first = 1;
+	  
+	  knot_desc desc_1;
+	  ptr<const module<Z2> > H_1;
+	  map<grading, basedvector<int, 1> > st_1;
+	  
+	  for (unsigned k = 1; k <= groups[j].size (); k ++)
+	    {
+	      knot_desc desc (knot_desc::HTW, i, groups[j][k]);
+	      
+	      if (mutant_knot_sq % desc)
+		{
+		  pair<mod_map<Z2>, mod_map<Z2> > p = mutant_knot_sq(desc);
+		  
+		  mod_map<Z2> sq1 = p.first,
+		    sq2 = p.second;
+		  ptr<const module<Z2> > H = sq1.domain ();
+		  
+		  map<grading, basedvector<int, 1> > st = compute_st (sq1, sq2);
+		  
+		  if (first)
+		    {
+		      first = 0;
+		      desc_1 = desc;
+		      H_1 = H;
+		      st_1 = st;
+		    }
+		  else
+		    {
+		      if (H_1->free_poincare_polynomial ()
+			  == H->free_poincare_polynomial ())
+			{
+			  if (st_1 != st)
+			    {
+			      printf ("> DIFFER\n");
+			      display ("  ", desc_1);
+			      display ("  ", desc);
+			      fflush (stdout);
+			    }
+			  compared ++;
+			}
+		      else if (mutant_mknot_sq % desc)
+			{
+			  pair<mod_map<Z2>, mod_map<Z2> > mp = mutant_mknot_sq(desc);
+			  
+			  mod_map<Z2> msq1 = mp.first,
+			    msq2 = mp.second;
+			  ptr<const module<Z2> > mH = msq1.domain ();
+			  
+			  assert (H_1->free_poincare_polynomial ()
+				  == mH->free_poincare_polynomial ());
+			  
+			  map<grading, basedvector<int, 1> > mst
+			    = compute_st (msq1, msq2);
+			  
+			  if (st_1 != mst)
+			    {
+			      printf ("> DIFFER\n");
+			      display ("  ", desc_1);
+			      display ("  ", desc);
+			      fflush (stdout);
+			    }
+			  compared ++;
+			}
+		      else
+			missing ++;
+		    }
+		}
+	      else
+		missing ++;
+	    }
+	}
+    }
+  printf ("missing = %d\n", missing);
+  printf ("compared = %d\n", compared);
+  fflush (stdout);
+}
+
+bool
+file_exists (const char *file)
+{
+  struct stat stat_buf;
+  if (stat (file, &stat_buf) != 0)
+    {
+      if (errno == ENOENT)
+	return 0;
+      
+      stderror ("stat: %s", file);
+      exit (EXIT_FAILURE);
+    }
+  
+  return 1;
+}
+
+#if 0
+bool
+file_exists (const char *buf)
+{
+  struct stat stat_buf;
+  if (stat (buf, &stat_buf) != 0)
+    {
+      if (errno == ENOENT)
+	return 0;
+      
+      stderror ("stat: %s", buf);
+      exit (EXIT_FAILURE);
+    }
+  
+  return 1;
+}
+
+void
+convert_mut15 ()
+{
+  hashmap<knot_desc,
+	  pair<mod_map<Z2>, mod_map<Z2> > > knot15_sq_part;
+  
+  basedvector<basedvector<unsigned, 1>, 1> groups = mutant_knot_groups (15);
+  for (unsigned i = 1; i <= groups.size (); i ++)
+    {
+      for (unsigned j = 1; j <= groups[i].size (); j ++)
+	{
+	  knot_desc desc (knot_desc::HTW, i, groups[i][j]);
+	  
+	  char buf[1000];
+	  sprintf (buf, "/u/cseed/mut15.bak/K15_%d.dat.gz", groups[i][j]);
+	  
+	  if (file_exists (buf))
+	    {
+	      gzfile_reader r (buf);
+	      
+	      mod_map<Z2> sq1 (r);
+	      mod_map<Z2> sq2 (r);
+	      knot15_sq_part.push (desc,
+				   pair<mod_map<Z2>, mod_map<Z2> > (sq1, sq2));
+	    }
+	}
+    }
+  
+  printf ("|knot15_sq_part| = %d\n", knot15_sq_part.card ());
+  
+  gzfile_writer w ("/u/cseed/src/knotkit/knot_sq/K15_sq_part.dat.gz");
+  write (w, knot15_sq_part);
+}
+#endif
+
+void
+compute_mutant_mirrors ()
+{
+  const char *knot_sq_files[] = {
+    // "knot_sq/K11_sq.dat.gz",
+    // "knot_sq/K12_sq.dat.gz",
+    // "knot_sq/K13_sq.dat.gz",
+    "knot_sq/K14_sq.dat.gz",
+    0,
+  };
+  unsigned minn = 14,
+    maxn = 14;
+  
+  hashset<knot_desc> mutants;
+  for (unsigned i = minn; i <= maxn; i ++)
+    {
+      basedvector<basedvector<unsigned, 1>, 1> groups
+	= mutant_knot_groups (i);
+      for (unsigned j = 1; j <= groups.size (); j ++)
+	{
+	  for (unsigned k = 1; k <= groups[j].size (); k ++)
+	    mutants += knot_desc (knot_desc::HTW, i, groups[j][k]);
+	}
+    }
+  printf ("|mutants| = %d\n", mutants.card ());
+  fflush (stdout);
+  
+  hashmap<knot_desc,
+	  pair<mod_map<Z2>, mod_map<Z2> > > mutant_knot_sq;
+  
+  for (unsigned i = 0; knot_sq_files[i]; i ++)
+    {
+      gzfile_reader r (knot_sq_files[i]);
+      
+      printf ("loading %s...\n", knot_sq_files[i]);
+      fflush (stdout);
+      
+      unsigned n = r.read_unsigned ();
+      for (unsigned i = 1; i <= n; i ++)
+	{
+	  knot_desc k (r);
+	  
+	  pair<mod_map<Z2>, mod_map<Z2> > p (r);
+	  if (mutants % k)
+	    mutant_knot_sq.push (k, p);
+	}
+    }
+  
+  printf ("|mutant_knot_sq| = %d\n", mutant_knot_sq.card ());
+  fflush (stdout);
+  
+  map<knot_desc,
+      pair<mod_map<Z2>, mod_map<Z2> > > mknot_kh_sq;
+  for (unsigned i = minn; i <= maxn; i ++)
+    {
+      basedvector<basedvector<unsigned, 1>, 1> groups
+	= mutant_knot_groups (i);
+      for (unsigned j = 1; j <= groups.size (); j ++)
+	{
+	  unsigned n = groups[j].size ();
+	  
+	  basedvector<ptr<const module<Z2> >, 1> group_H (n);
+	  for (unsigned k = 1; k <= n; k ++)
+	    {
+	      knot_desc desc (knot_desc::HTW, i, groups[j][k]);
+	      pair<mod_map<Z2>, mod_map<Z2> > p = mutant_knot_sq(desc);
+	      group_H[k] = p.first.domain ();
+	    }
+	  
+	  for (unsigned k = 2; k <= n; k ++)
+	    {
+	      if (group_H[1]->free_poincare_polynomial ()
+		  != group_H[k]->free_poincare_polynomial ())
+		{
+		  knot_desc desc (knot_desc::HTW, i, groups[j][k]);
+		  show (desc); newline ();
+		  fflush (stdout);
+		  
+		  char buf[1000];
+		  assert (desc.t == knot_desc::HTW);
+		  sprintf (buf, "mknot/K%d_%d.dat.gz", desc.i, desc.j);
+		  
+		  if (!file_exists (buf))
+		    {
+		      knot_diagram kd (MIRROR, desc.diagram ());
+		  
+		      cube<Z2> c (kd);
+		      mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
+		  
+		      chain_complex_simplifier<Z2> s (c.khC, d, 1);
+		      assert (s.new_d == 0);
+		  
+		      steenrod_square sq (c, d, s);
+		      mod_map<Z2> sq1 = sq.sq1 ();
+		      mod_map<Z2> sq2 = sq.sq2 ();
+		  
+		      ptr<const module<Z2> > H = sq1.domain ();
+		  
+		      assert (group_H[1]->free_poincare_polynomial ()
+			      == H->free_poincare_polynomial ());
+		  
+		      gzfile_writer w (buf);
+		      write (w, desc);
+		      write (w, sq1);
+		      write (w, sq2);
+		    }
+		}
+	    }
+	}
+    }
+}
+
+void
+convert_mknots ()
+{
+  hashmap<knot_desc,
+	  pair<mod_map<Z2>, mod_map<Z2> > > mutant_mknot_sq;
+  
+  for (unsigned i = 11; i <= 15; i ++)
+    {
+      basedvector<basedvector<unsigned, 1>, 1> groups
+	= mutant_knot_groups (i);
+      for (unsigned j = 1; j <= groups.size (); j ++)
+	{
+	  unsigned n = groups[j].size ();
+	  for (unsigned k = 1; k <= n; k ++)
+	    {
+	      knot_desc desc (knot_desc::HTW, i, groups[j][k]);
+	      
+	      char buf[1000];
+	      assert (desc.t == knot_desc::HTW);
+	      sprintf (buf, "mknot/K%d_%d.dat.gz", desc.i, desc.j);
+	      
+	      struct stat stat_buf;
+	      if (stat (buf, &stat_buf) != 0)
+		{
+		  if (errno != ENOENT)
+		    {
+		      stderror ("stat: %s", buf);
+		      exit (EXIT_FAILURE);
+		    }		      
+		}
+	      else
+		{
+		  printf ("loading %s...\n", buf);
+		  
+		  gzfile_reader r (buf);
+		  
+		  knot_desc desc_2 (r);
+		  assert (desc_2 == desc);
+		  
+		  mod_map<Z2> sq1 (r);
+		  mod_map<Z2> sq2 (r);
+
+		  mutant_mknot_sq.push (desc,
+					pair<mod_map<Z2>, mod_map<Z2> > (sq1, sq2));
+		}
+	    }
+	}
+    }
+  
+  gzfile_writer w ("knot_sq/mut_mknot.dat.gz");
+  write (w, mutant_mknot_sq);
+}
+
+void
+convert_15 ()
+{
+  hashmap<knot_desc,
+    pair<mod_map<Z2>, mod_map<Z2> > > knot15_sq;
+  
+  basedvector<basedvector<unsigned, 1>, 1> groups
+    = mutant_knot_groups (15);
+  for (unsigned i = 1; i <= groups.size (); i ++)
+    {
+      for (unsigned j = 1; j <= groups[i].size (); j ++)
+	{
+	  unsigned k = groups[i][j];
+	  
+	  char buf[1000];
+	  sprintf (buf, "/u/cseed/mut15.bak/K15_%d.dat.gz", k);
+	  
+	  if (file_exists (buf))
+	    {
+	      printf ("loading %s...\n", buf);
+	      fflush (stdout);
+	      
+	      gzfile_reader r (buf);
+	      mod_map<Z2> sq1 (r);
+	      mod_map<Z2> sq2 (r);
+	      
+	      knot_desc desc (knot_desc::HTW, 15, k);
+	      knot15_sq.push (desc,
+			      pair<mod_map<Z2>, mod_map<Z2> > (sq1, sq2));
+	    }
+	}
+    }
+  
+  printf ("|knot15_sq| = %d\n", knot15_sq.card ());
+  
+  gzfile_writer w ("/u/cseed/src/knotkit/K15_sq_part.dat.gz");
+  write (w, knot15_sq);
+}
+
+void
+compare_gss_splitting ()
+{
+  typedef Z2 R;
+  
+#if 1
+  for (unsigned i = 1; i <= 14; i ++)
+    for (unsigned j = 1; j <= mt_links (i, 1); j ++)
+      {
+	knot_diagram kd (mt_link (i, 1, j));
+#endif
+	
+	unsigned n = kd.num_components ();
+	if (n < 2)
+	  continue;
+	
+	show (kd); newline ();
+	
+	unionfind<1> u (kd.num_edges ());
+	for (unsigned i = 1; i <= kd.n_crossings; i ++)
+	  {
+	    u.join (kd.ept_edge (kd.crossings[i][1]),
+		    kd.ept_edge (kd.crossings[i][3]));
+	    u.join (kd.ept_edge (kd.crossings[i][2]),
+		    kd.ept_edge (kd.crossings[i][4]));
+	  }
+	
+	map<unsigned, unsigned> root_comp;
+	unsigned t = 0;
+	for (unsigned i = 1; i <= kd.num_edges (); i ++)
+	  {
+	    if (u.find (i) == i)
+	      {
+		++ t;
+		root_comp.push (i, t);
+	      }
+	  }
+	assert (t == n);
+	
+	cube<R> c (kd);
+#if 1
+	for (unsigned i = 0; i < c.n_resolutions; i ++)
+	  {
+	    smallbitset state (kd.n_crossings, i);
+	    smoothing s (kd, state);
+	    s.show_self (kd, state);
+	    newline ();
+	  }
+#endif
+
+	printf ("inter-component x:\n");
+	mod_map<R> ds (c.khC);
+	for (unsigned x = 1; x <= kd.n_crossings; x ++)
+	  {
+	    unsigned p1 = kd.crossings[x][1],
+	      p2 = kd.crossings[x][2];
+	    unsigned r1 = u.find (kd.ept_edge (p1)),
+	      r2 = u.find (kd.ept_edge (p2));
+	    unsigned c1 = root_comp(r1),
+	      c2 = root_comp(r2);
+
+	    if (c1 != c2)
+	      {
+		printf ("  %d\n", x);
+		
+		ds = ds + c.compute_dinv (x);
+	      }
+	  }
+	
+	mod_map<R> d1 = c.compute_d (1, 0, 0, 0, 0);
+	assert (d1.compose (ds) == ds.compose (d1));
+
+	mod_map<R> d2 = c.compute_d (2, 0, 0, 0, 0);
+	mod_map<R> h2 = d2.compose (ds) + ds.compose (d2);
+	if (h2 != 0)
+	  display ("h2:\n", h2);
+	
+	mod_map<R> d3 = c.compute_d (3, 0, 0, 0, 0);
+	mod_map<R> h3 = d3.compose (ds) + ds.compose (d3);
+	if (h3 != 0)
+	  display ("h3:\n", h3);
+	
+	mod_map<R> d = c.compute_d (0, 0, 0, 0, 0);
+	mod_map<R> h = d.compose (ds) + ds.compose (d);
+	if (h != 0)
+	  display ("h:\n", h);
+      }
+}
+
+template<class R> unsigned
+splitting_bound (knot_diagram &kd,
+		 basedvector<R, 1> comp_weight)
+{
+  sseq ss = compute_forgetfulss<R> (kd, comp_weight);
+  
+  assert (ss.pages.size () >= 1);
+  return ss.pages.size () - 1;
+}
+
+basedvector<basedvector<unsigned, 1>, 1>
+permutations (basedvector<unsigned, 1> v)
+{
+  unsigned n = v.size ();
+  basedvector<basedvector<unsigned, 1>, 1> ps;
+  
+  if (n == 1)
+    {
+      ps.append (v);
+      return ps;
+    }
+  
+  for (unsigned i = 1; i <= n; i ++)
+    {
+      unsigned x = v[i];
+      
+      basedvector<unsigned, 1> v2 (n - 1);
+      for (unsigned j = 1; j < i; j ++)
+	v2[j] = v[j];
+      for (unsigned j = i + 1; j <= n; j ++)
+	v2[j - 1] = v[j];
+      
+      basedvector<basedvector<unsigned, 1>, 1> ps2 = permutations (v2);
+      for (unsigned j = 1; j <= ps2.size (); j ++)
+	{
+	  basedvector<unsigned, 1> p2 = ps2[j];
+	  assert (p2.size () == n - 1);
+	  
+	  basedvector<unsigned, 1> v3 (n);
+	  v3[1] = x;
+	  for (unsigned k = 1; k <= n - 1; k ++)
+	    v3[k + 1] = p2[k];
+	  ps.append (v3);
+	}
+    }
+  
+  return ps;
+}
+
+basedvector<basedvector<unsigned, 1>, 1>
+permutations (unsigned n)
+{
+  basedvector<unsigned, 1> v (n);
+  for (unsigned i = 1; i <= n; i ++)
+    v[i] = i;
+  return permutations (v);
+}
+
+template<class R> multivariate_laurentpoly<Z>
+Kh_poincare_polynomial (knot_diagram &kd)
+{
+  cube<R> c (kd);
+  mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
+  chain_complex_simplifier<R> s (c.khC, d, 0);
+  assert (s.new_d == 0);
+  return s.new_C->free_poincare_polynomial ();
+}
+
+unsigned
+compute_b_lk_weak (knot_diagram &kd)
+{
+  unsigned m = kd.num_components ();
+  assert (m > 1);
+  
+  if (m == 2)
+    {
+      unsigned total_lk = kd.total_linking_number ();
+      return total_lk == 0 ? 2 : total_lk;
+    }
+  
+  unionfind<1> u (kd.num_edges ());
+  
+  for (unsigned i = 1; i <= kd.n_crossings; i ++)
+    {
+      u.join (kd.ept_edge (kd.crossings[i][1]),
+	      kd.ept_edge (kd.crossings[i][3]));
+      u.join (kd.ept_edge (kd.crossings[i][2]),
+	      kd.ept_edge (kd.crossings[i][4]));
+    }
+  assert (m == u.num_sets ());
+  
+  map<unsigned, unsigned> root_comp;
+  unsigned t = 0;
+  for (unsigned i = 1; i <= kd.num_edges (); i ++)
+    {
+      if (u.find (i) == i)
+	{
+	  ++ t;
+	  root_comp.push (i, t);
+	}
+    }
+  assert (t == m);
+  
+  unsigned b_lk_weak = 0;
+  
+  for (unsigned i = 1; i <= m; i ++)
+    for (unsigned j = i + 1; j <= m; j ++)
+      {
+	assert (i < j);
+	
+	int lk = 0;
+	for (unsigned x = 1; x <= kd.n_crossings; x ++)
+	  {
+	    unsigned r1 = root_comp(u.find (kd.ept_edge (kd.crossings[x][1]))),
+	      r2 = root_comp(u.find (kd.ept_edge (kd.crossings[x][2])));
+	    if (((r1 == i) && (r2 == j))
+		|| ((r2 == i) && (r1 == j)))
+	      {
+		if (kd.is_to_ept (kd.crossings[x][1]) == kd.is_to_ept (kd.crossings[x][4]))
+		  lk ++;
+		else
+		  lk --;
+	      }
+	  }
+	assert (is_even (lk));
+	lk /= 2;
+	
+	if (lk == 0)
+	  {
+	    smallbitset ci (m);
+	    ci.push (i);
+	    
+	    smallbitset cj (m);
+	    cj.push (j);
+	    
+	    smallbitset c (m);
+	    c.push (i);
+	    c.push (j);
+	    knot_diagram Lij (SUBLINK, c, kd);
+	    
+	    multivariate_laurentpoly<Z> P_Lij = Kh_poincare_polynomial<Z2> (Lij);
+	    
+	    knot_diagram Li_join_Lj (DISJOINT_UNION,
+				     knot_diagram (SUBLINK, ci, kd),
+				     knot_diagram (SUBLINK, cj, kd));
+	
+	    multivariate_laurentpoly<Z> P_Li_join_Lj = Kh_poincare_polynomial<Z2> (Li_join_Lj);
+	    
+	    if (P_Lij != P_Li_join_Lj)
+	      lk = 2;  // non-split
+	  }
+	
+	b_lk_weak += abs (lk);
+      }
+  
+  return b_lk_weak == 0 ? 2 : b_lk_weak;
+}
+
+void
+compute_splitting_bounds ()
+{
+  typedef fraction_field<polynomial<Z2> > Z2x;
+  
   for (unsigned i = 1; i <= 12; i ++)
+    for (unsigned j = 1; j <= mt_links (i); j ++)
+      {
+	knot_diagram kd (mt_link (i, j));
+	unsigned m = kd.num_components ();
+	if (m == 1)
+	  continue;
+	
+	show (kd); newline ();
+	printf ("  m = %d\n", m);
+	
+	unionfind<1> u (kd.num_edges ());
+	
+	for (unsigned i = 1; i <= kd.n_crossings; i ++)
+	  {
+	    u.join (kd.ept_edge (kd.crossings[i][1]),
+		    kd.ept_edge (kd.crossings[i][3]));
+	    u.join (kd.ept_edge (kd.crossings[i][2]),
+		    kd.ept_edge (kd.crossings[i][4]));
+	  }
+	assert (u.num_sets () == m);
+	
+	map<unsigned, unsigned> root_comp;
+	unsigned t = 0;
+	for (unsigned i = 1; i <= kd.num_edges (); i ++)
+	  {
+	    if (u.find (i) == i)
+	      {
+		++ t;
+		root_comp.push (i, t);
+	      }
+	  }
+	assert (t == m);
+	
+	basedvector<Q, 1> comp_weightQ (m);
+	for (unsigned i = 1; i <= m; i ++)
+	  comp_weightQ[i] = Q (i);
+	unsigned bQ = splitting_bound<Q> (kd, comp_weightQ);
+	
+	basedvector<Z2x, 1> comp_weightZ2x (m);
+	for (unsigned i = 1; i <= m; i ++)
+	  comp_weightZ2x[i] = Z2x (polynomial<Z2> (Z2 (1), i));
+	unsigned bZ2x = splitting_bound<Z2x> (kd, comp_weightZ2x);
+	
+	// lower bound
+	unsigned b = std::max (bQ, bZ2x);
+	
+	printf ("  bQ = %d\n", bQ);
+	printf ("  bZ2x = %d\n", bZ2x);
+	printf ("  b = %d\n", b);
+	
+	unsigned total_lk = kd.total_linking_number ();
+	
+	unsigned b_lk_weak = compute_b_lk_weak (kd);
+	unsigned b_lk_weaker = total_lk == 0 ? 2 : total_lk;
+	
+	printf ("  b_lk_weaker = %d\n", b_lk_weaker);
+	printf ("  b_lk_weak = %d\n", b_lk_weak);
+	
+	assert (b_lk_weaker <= b_lk_weak);
+	if (b_lk_weaker < b_lk_weak)
+	  printf (" > STRICTLY WEAKER\n");
+	
+	basedvector<basedvector<unsigned, 1>, 1> ps = permutations (m);
+	unsigned r = kd.n_crossings;
+	for (unsigned i = 1; i <= ps.size (); i ++)
+	  {
+	    basedvector<unsigned, 1> p = ps[i];
+	    
+	    unsigned ri = 0;
+	    for (unsigned j = 1; j <= kd.n_crossings; j ++)
+	      {
+		unsigned upper_e = kd.ept_edge (kd.crossings[j][2]),
+		  lower_e = kd.ept_edge (kd.crossings[j][1]);
+		
+		unsigned upper_c = root_comp(u.find (upper_e)),
+		  lower_c = root_comp(u.find (lower_e));
+		
+		if (upper_c != lower_c
+		    && p[upper_c] < p[lower_c])
+		  ri ++;
+	      }
+	    
+	    if (ri < r)
+	      r = ri;
+	  }
+	printf ("  r = %d\n", r);
+	
+	assert (b_lk_weak <= r);
+	assert (b <= r);
+	
+	// non-trivial link, sp at least 1.
+	assert (b_lk_weak >= 1);
+	if (b <= 1
+	    && b_lk_weak == 1)
+	  continue;
+	
+	if (b > b_lk_weak)
+	  {
+	    if (b == r)
+	      printf ("  > sp = %d (b)\n", b);
+	    else
+	      printf ("  > %d <= sp <= %d (b)\n", b, r);
+	  }
+	else if (b == b_lk_weak)
+	  {
+	    if (b == r)
+	      printf ("  > sp = %d (b + b_lk_weak)\n", b);
+	    else
+	      printf ("  > %d <= sp <= %d (b + b_lk_weak)\n", b, r);
+	  }
+	else if (b_lk_weak == r)
+	  {
+	    assert (b < b_lk_weak);
+	    printf ("  > sp = %d (b_lk_weak)\n", b_lk_weak);
+	  }
+      }
+}
+
+
+void
+compute_forgetful_tables ()
+{
+  // typedef fraction_field<multivariate_polynomial<Z2, 10> > R;
+  typedef Z2 R;
+  
+#if 0
+  for (unsigned i = 1; i <= 10; i ++)
+    for (unsigned j = 1; j <= mt_links (i); j ++)
+      {
+	knot_diagram kd (mt_link (i, j));
+#endif
+      {
+	// knot_diagram kd (mt_link (12, 0, 2087));
+	// knot_diagram kd (mt_link (12, 0, 1705));
+	// knot_diagram kd (mt_link (14, 0, 66759));
+	// knot_diagram kd (mt_link (14, 0, 65798));
+	knot_diagram kd (mt_link (13, 0, 8862));
+	
+	abort ();
+	// sseq ss = compute_forgetfulss<R> (kd);
+	sseq ss;
+	
+#if 0
+	if (ss.pages.size () < 2)
+	  continue;
+#endif
+	
+	for (unsigned k = 1; k <= ss.pages.size (); k ++)
+	  {
+	    if (k == 1)
+	      printf ("%s &", kd.name.c_str ());
+	    else
+	      printf (" &");
+	    
+	    printf (" $E_%d$ & %d & $", k, ss.pages[k].total_rank ());
+	    
+	    bool first = 1;
+	    
+	    const sseq_bounds &b = ss.bounds;
+	    
+	    for (int i = b.minh; i <= b.maxh; i ++)
+	      for (int j = b.minq; j <= b.maxq; j ++)
+		{
+		  unsigned r = ss.pages[k].rank[i - b.minh][j - b.minq];
+		  
+		  if (r > 0)
+		    {
+		      
+		      if (first)
+			first = 0;
+		      else
+			printf (" + ");
+		
+		      if (i == 0
+			  && j == 0)
+			printf ("%d", r);
+		      else
+			{
+			  if (r != 1)
+			    printf ("%d", r);
+		    
+			  if (i != 0)
+			    printf ("t^{%d}", i);
+			  if (j != 0)
+			    printf ("q^{%d}", j);
+			}
+		    }
+		}
+	    printf ("$ \\\\\n");
+	  }
+      }
+}
+
+void
+show_lk (const dt_code &dt)
+{
+  knot_diagram kd (dt);
+  
+  show (kd); newline ();
+  printf ("m = %d\n", kd.num_components ());
+  
+  planar_diagram (kd).display_knottheory ();
+}
+    
+int
+main ()
+{
+  show_lk (mt_link (11, 1, 739));
+  show_lk (mt_link (12, 1, 2521));
+  show_lk (mt_link (12, 1, 2552));
+  show_lk (mt_link (12, 1, 2672));
+  show_lk (mt_link (12, 1, 2910));
+  
+  show_lk (mt_link (12, 0, 2207));
+  show_lk (mt_link (12, 0, 2209));
+  show_lk (mt_link (12, 0, 2211));
+  show_lk (mt_link (12, 0, 2214));
+  show_lk (mt_link (12, 0, 2208));
+  show_lk (mt_link (12, 0, 2230));
+  show_lk (mt_link (12, 0, 2238));
+  show_lk (mt_link (12, 0, 2245));
+  show_lk (mt_link (12, 0, 2251));
+  show_lk (mt_link (12, 0, 2255));
+  show_lk (mt_link (12, 0, 2262));
+  show_lk (mt_link (12, 0, 2292));
+  return 0;
+  
+  {
+    knot_diagram kd (mt_link (12, 0, 2262));
+    printf ("m = %d\n", kd.num_components ());
+    
+    planar_diagram (kd).display_knottheory ();
+  }
+  return 0;
+  
+  compute_splitting_bounds ();
+  return 0;
+  
+  compute_forgetful_tables ();
+  return 0;
+  
+#if 0
+  {
+    knot_diagram kd (mt_link (5, 1, 3));
+    show (kd); newline ();
+
+    sseq ss = compute_forgetfulss<Q> (kd);
+    ss.texshow (stdout, "L5a3");
+  }
+  return 0;
+#endif
+  
+  compute_lee_bound ();
+  return 0;
+  
+  compare_gss_splitting ();
+  
+  {
+    knot_diagram kd (rolfsen_knot (3, 1));
+    cube<Q> c (kd);
+    mod_map<Q> d = c.compute_d (1, 0, 0, 0, 0);
+    chain_complex_simplifier<Q> s (c.khC, d, 0);
+    assert (s.new_d == 0);
+    display ("s.new_C:\n", *s.new_C);
+  }
+  return 0;
+  
+  test_forgetful_signs ();
+  
+  {
+    knot_diagram kd (torus_knot (2, 2));
+    printf ("n+ = %d, n- = %d\n", kd.nplus, kd.nminus);
+    cube<Z2> c (kd);
+    sseq ss = compute_szabo_sseq (c);
+    ss.texshow (stdout, "Hopf");
+  }
+  {
+    knot_diagram kd (MIRROR, knot_diagram (torus_knot (2, 2)));
+    printf ("n+ = %d, n- = %d\n", kd.nplus, kd.nminus);
+    cube<Z2> c (kd);
+    sseq ss = compute_szabo_sseq (c);
+    ss.texshow (stdout, "Hopf");
+  }
+  
+  return 0;
+  
+  compute_forgetful_torsion ();
+  return 0;
+  
+  test_forgetful_signs ();
+  test_crossing_ss ();
+  
+  find_width4_in_h0 ();
+  // convert_15 ();
+  // test_knot_sq ();
+  return 0;
+  
+  // convert_mut15 ();
+  
+  compute_mutant_mirrors ();
+  
+  test_kh_htt_stronger ();
+  
+  compute_mutant_mirrors ();
+  
+  test_knot_sq ();
+  
+  convert_mknots ();
+  
+  test_forgetful_ss ();
+  compute_twistedU ();
+  
+#if 1
+  knot_desc desc;
+  desc.t = knot_desc::HTW;
+  desc.i = 15;
+  desc.j = 15020;
+  
+  char buf[1000];
+  sprintf (buf, "/scratch/network/cseed/incoming/K%d_%d.dat.gz",
+	   desc.i, desc.j);
+  
+  desc = knot_desc (knot_desc::ROLFSEN, 3, 1);
+
+  knot_diagram kd = desc.diagram ();
+
+  cube<Z2> c (kd);
+  mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
+  
+  chain_complex_simplifier<Z2> s (c.khC, d, 1);
+  assert (s.new_d == 0);
+  
+  steenrod_square sq (c, d, s);
+  mod_map<Z2> sq1 = sq.sq1 ();
+  mod_map<Z2> sq2 = sq.sq2 ();
+  
+  assert (sq1.compose (sq1) == 0);
+  assert (sq2.compose (sq2) + sq1.compose (sq2).compose (sq1) == 0);
+
+#if 1
+  {
+    file_writer w (buf);
+    write (w, sq1);
+    write (w, sq2);
+  }
+#endif
+#endif
+  return 0;
+  
+  // convert_knot_sq ();
+  test_knot_sq ();
+  
+  // test_sage_show ();
+  dump_sage ();
+  
+#if 0
+  knot_diagram kd (rolfsen_knot (8, 19));
+  show (kd); newline ();
+  
+  cube<Z2> c (kd);
+  mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
+  
+  chain_complex_simplifier<Z2> s (c.khC, d, 1);
+  
+  steenrod_square sq (c, d, s);
+  mod_map<Z2> sq1 = sq.sq1 ();
+  mod_map<Z2> sq2 = sq.sq2 ();
+  
+  assert (sq1.compose (sq1) == 0);
+  assert (sq2.compose (sq2) + sq1.compose (sq2).compose (sq1) == 0);
+  
+  display ("sq1:\n", sq1);
+  display ("sq2:\n", sq2);
+  
+  {
+    file_writer w ("sqtest.dat");
+    write (w, sq1);
+    write (w, sq2);
+  }
+  
+  {
+    file_reader r ("sqtest.dat");
+    mod_map<Z2> sq1p (r);
+    mod_map<Z2> sq2p (r);
+    display ("sq1p:\n", sq1p);
+    display ("sq2p:\n", sq2p);
+  }
+  
+#endif
+
+#if 0
+  knot_diagram kd (mt_link (10, 0, 9));
+  cube<Z2> c (kd);
+#endif
+  
+#if 0
+  for (unsigned i = 1; i <= 10; i ++)
+    for (unsigned j = 1; j <= mt_links (i, 0); j ++)
+      {
+	knot_diagram kd (mt_link (i, 0, j));
+	kd.marked_edge = 1;
+	
+	cube<Z2> c (kd, 1);
+	sseq ss = compute_szabo_sseq (c);
+	ss.texshow (stdout, kd.name);
+      }
+#endif
+  
+#if 0
+#if 1
+  for (unsigned i = 10; i <= 10; i ++)
+    for (unsigned j = 124; j <= rolfsen_crossing_knots (i); j ++)
+      {
+	knot_diagram kd (rolfsen_knot (i, j));
+#endif
+#if 0
+  for (unsigned i = 1; i <= 10; i ++)
+    for (unsigned j = 1; j <= mt_links (i, 0); j ++)
+      {
+#endif
+#if 0
+  for (unsigned i = 11; i <= 11; i ++)
+    for (unsigned j = 1; j <= htw_knots (i, 0); j ++)
+      {
+#endif
+	// knot_diagram kd (htw_knot (i, 0, j));
+	// knot_diagram kd (mt_link (i, 0, j));
+	kd.marked_edge = 1;
+	
+	show (kd); newline ();
+  
+	cube<Z2> c (kd, 1);
+	
+#if 0
+	mod_map<Z2> d1 = c.compute_d (1, 0, 0, 0, 0);
+  
+	chain_complex_simplifier<Z2> s (c.khC, d1, 1);
+	assert (s.new_d == 0);
+#endif
+  
+	sseq ss = compute_szabo_sseq (c);
+  
+	multivariate_laurentpoly<Z> Phat = 
+	  ss.pages[ss.pages.size ()].delta_poincare_polynomial (ss.bounds);
+	
+	typedef spanning_tree_complex<Z2>::R R;
+	
+	spanning_tree_complex<Z2> spanc (kd);
+	mod_map<R> d2 = spanc.twisted_d2 ();
+	mod_map<R> d2U = spanc.twisted_d2Un (1);
+	
+	chain_complex_simplifier<R> s2 (spanc.C, d2, 2);
+	assert (s2.new_d == 0);
+	
+	mod_map<R> H_d2U = s2.pi.compose (d2U).compose (s2.iota);
+	assert (H_d2U.compose (H_d2U) == 0);
+	
+	ptr<const module<R> > ker = H_d2U.kernel ();
+	ptr<const module<R> > quot = s2.new_C->quotient (H_d2U.image ());
+        
+        multivariate_laurentpoly<Z> Pminus1
+	    = ker->free_delta_poincare_polynomial (),
+	  PminusU = quot->free_delta_poincare_polynomial ();
+	
+	if (PminusU != Pminus1)
+	  {
+	    display ("  HFhat: ", Phat);
+	    // display ("  HF-: ", Pminus);
+	    display ("  HF- (1): ", Pminus1);
+	    display ("  HF- (U): ", PminusU);
+	  }
+      }
+#endif
+  
+#if 0
+  hwidth_knots = basedvector<unsigned, 1> (10);
+  for (unsigned i = 1; i <= hwidth_knots.size (); i ++)
+    hwidth_knots[i] = 0;
+  
+  map<knot_desc,
+      pair<mod_map<Z2>, mod_map<Z2> > > knot_kh_sq;
+  
+  for (unsigned i = 14; i >= 1; i --)
+    {
+#if 0
+      if (i <= 10)
+	{
+	  for (unsigned j = 1; j <= rolfsen_crossing_knots (i); j += block_size)
+	    {
+	      load (knot_kh_sq, knot_desc (knot_desc::ROLFSEN, i, j));
+	    }
+	}
+#endif
+      
+      for (unsigned j = 1; j <= htw_knots (i); j += block_size)
+	{
+	  load (knot_kh_sq, knot_desc (knot_desc::HTW, i, j));
+	}
+
+      if (i <= 13)
+	{
+	  for (unsigned j = 1; j <= mt_links (i); j += block_size)
+	    {
+	      load (knot_kh_sq, knot_desc (knot_desc::MT, i, j));
+	    }
+	}
+      
+      if (i == 14)
+	{
+	  for (unsigned j = 1; j <= mt_links (14, 0); j += block_size)
+	    {
+	      load (knot_kh_sq, knot_desc (knot_desc::MT, 14, mt_links (14, 1) + j));
+	    }
+	}
+    }
+  
+  unsigned total_knots = 0;
+  printf ("hwidth_knots:\n");
+  for (unsigned i = 1; i <= hwidth_knots.size (); i ++)
+    {
+      printf ("  % 2d: %d\n", i, hwidth_knots[i]);
+      total_knots += hwidth_knots[i];
+    }
+  printf ("total_knots = %d\n", total_knots);
+  
+  printf ("|knot_kh_sq| = %d\n", knot_kh_sq.card ());
+  
+#if 0
+  map<pair<multivariate_laurentpoly<Z>,
+	   map<grading, unsigned> >,
+      pair<knot_desc,
+	   map<grading, basedvector<int, 1> > > > P_sq1_knot_st;
+  
+  set<multivariate_laurentpoly<Z> > Ps;
+  basedvector<unsigned, 1> collisons (10);
+  for (unsigned i = 1; i <= 10; i ++)
+    collisons[i] = 0;
+  
+  for (map<knot_desc,
+	   pair<mod_map<Z2>, mod_map<Z2> > >::const_iter i = knot_kh_sq; i; i ++)
+    {
+      show_st (knot_kh_sq, i.key ());
+      
+      mod_map<Z2> sq1 = i.val ().first;
+      mod_map<Z2> sq2 = i.val ().second;
+      
+#if 0
+      display ("sq1:\n", sq1);
+      display ("sq2:\n", sq2);
+#endif
+      
+      printf ("%s  ", i.key ().name ().c_str ());
+      
+      assert (sq1.compose (sq1) == 0);
+      assert (sq2.compose (sq2) + sq1.compose (sq2).compose (sq1) == 0);
+      
+      ptr<const module<Z2> > H = sq1.domain ();
+      unsigned hwidth = homological_width (H);
+      
+      map<grading, basedvector<int, 1> > st;
+      map<grading, unsigned> sq1_ranks;
+      
+      bool first = 1;
+      
+      set<grading> gs = H->gradings ();
+      
+      for (set_const_iter<grading> gg = gs; gg; gg ++)
+	{
+	  grading hq = gg.val (),
+	    h1q (hq.h + 1, hq.q),
+	    h2q (hq.h + 2, hq.q),
+	    h3q (hq.h + 3, hq.q);
+	  
+	  // printf ("(%d, %d):\n", hq.h, hq.q);
+	  
+	  ptr<const free_submodule<Z2> > H_hq = H->graded_piece (hq),
+	    H_h1q = H->graded_piece (h1q),
+	    H_h2q = H->graded_piece (h2q),
+	    H_h3q = H->graded_piece (h3q);
+	  
+	  mod_map<Z2> S = sq2.restrict (H_hq, H_h2q),
+	    T = sq2.restrict (H_h1q, H_h3q),
+	    A = sq1.restrict (H_hq, H_h1q),
+	    B = sq1.restrict (H_h1q, H_h2q),
+	    C = sq1.restrict (H_h2q, H_h3q);
+	  
+	  ptr<const free_submodule<Z2> > Sker = S.kernel (),
+	    Sim = S.image (),
+	    Tker = T.kernel (),
+	    Tim = T.image (),
+	    Aker = A.kernel (),
+	    Aim = A.image (),
+	    Bker = B.kernel (),
+	    Bim = B.image (),
+	    Cker = C.kernel (),
+	    Cim = C.image ();
+	  
+	  sq1_ranks.push (hq, Aim->dim ());
+	  
+	  mod_map<Z2> ArSker = A.restrict_from (Sker);
+	  mod_map<Z2> SrAker = S.restrict_from (Aker);
+	  
+	  mod_map<Z2> TrAim = T.restrict_from (Aim);
+	  mod_map<Z2> TrBker = T.restrict_from (Bker);
+	  mod_map<Z2> BrTker = B.restrict_from (Tker);
+	  
+	  mod_map<Z2> CrSim = C.restrict_from (Sim);
+	  
+	  ptr<const free_submodule<Z2> > ArSker_im = ArSker.image ();
+	  ptr<const free_submodule<Z2> > SrAker_im = SrAker.image ();
+	  
+	  ptr<const free_submodule<Z2> > TrAim_im = TrAim.image ();
+	  ptr<const free_submodule<Z2> > TrBker_im = TrBker.image ();
+	  ptr<const free_submodule<Z2> > BrTker_im = BrTker.image ();
+	  
+	  ptr<const free_submodule<Z2> > CrSim_im = CrSim.image ();
+	  
+	  mod_map<Z2> CrSrAker_im = C.restrict_from (SrAker_im);
+	  mod_map<Z2> TrArSker_im = T.restrict_from (ArSker_im);
+	  
+	  ptr<const free_submodule<Z2> > CrSrAker_im_im = CrSrAker_im.image ();
+	  ptr<const free_submodule<Z2> > TrArSker_im_im = TrArSker_im.image ();
+	  
+	  ptr<const free_submodule<Z2> > Aker_cap_Sker = Aker->intersection (Sker);
+	  
+	  ptr<const free_submodule<Z2> > Aim_cap_Tker = Aim->intersection (Tker);
+	  ptr<const free_submodule<Z2> > Bker_cap_Tker = Bker->intersection (Tker);
+	  ptr<const free_submodule<Z2> > ArSker_im_cap_Tker = ArSker_im->intersection (Tker);
+	  
+	  ptr<const free_submodule<Z2> > Sim_cap_Bim = Sim->intersection (Bim);
+	  ptr<const free_submodule<Z2> > Sim_cap_BrTker_im = Sim->intersection (BrTker_im);
+	  ptr<const free_submodule<Z2> > Sim_cap_Cker = Sim->intersection (Cker);
+	  ptr<const free_submodule<Z2> > SrAker_im_cap_Bim = SrAker_im->intersection (Bim);
+	  ptr<const free_submodule<Z2> > SrAker_im_cap_BrTker_im = SrAker_im->intersection (BrTker_im);
+	  ptr<const free_submodule<Z2> > SrAker_im_cap_Cker = SrAker_im->intersection (Cker);
+	  
+	  ptr<const free_submodule<Z2> > Tim_cap_Cim = Tim->intersection (Cim);
+	  ptr<const free_submodule<Z2> > TrAim_im_cap_Cim = TrAim_im->intersection (Cim);
+	  ptr<const free_submodule<Z2> > TrBker_im_cap_Cim = TrBker_im->intersection (Cim);
+	  ptr<const free_submodule<Z2> > TrArSker_im_im_cap_Cim = TrBker_im->intersection (Cim);
+	  
+	  ptr<const free_submodule<Z2> > Tim_cap_CrSim_im = Tim->intersection (CrSim_im);
+	  ptr<const free_submodule<Z2> > TrAim_im_cap_CrSim_im = TrAim_im->intersection (CrSim_im);
+	  ptr<const free_submodule<Z2> > TrBker_im_cap_CrSim_im = TrBker_im->intersection (CrSim_im);
+	  ptr<const free_submodule<Z2> > TrArSker_im_im_cap_CrSim_im = TrBker_im->intersection (CrSim_im);
+
+	  ptr<const free_submodule<Z2> > Tim_cap_CrSrAker_im_im = Tim->intersection (CrSrAker_im_im);
+	  ptr<const free_submodule<Z2> > TrAim_im_cap_CrSrAker_im_im = TrAim_im->intersection (CrSrAker_im_im);
+	  ptr<const free_submodule<Z2> > TrBker_im_cap_CrSrAker_im_im = TrBker_im->intersection (CrSrAker_im_im);
+	  ptr<const free_submodule<Z2> > TrArSker_im_im_cap_CrSrAker_im_im = TrBker_im->intersection (CrSrAker_im_im);
+	  
+	  basedvector<int, 1> v;
+	  
+	  v.append (Sker->dim ());
+	  v.append (Sim->dim ());
+	  v.append (Tker->dim ());
+	  v.append (Tim->dim ());
+	  
+	  v.append (Aker->dim ());
+	  v.append (Aim->dim ());
+	  v.append (Bker->dim ());
+	  v.append (Bim->dim ());
+	  v.append (Cker->dim ());
+	  v.append (Cim->dim ());
+	  
+	  v.append (ArSker_im->dim ());
+	  v.append (SrAker_im->dim ());
+	  
+	  v.append (TrAim_im->dim ());
+	  v.append (TrBker_im->dim ());
+	  v.append (BrTker_im->dim ());
+	  
+	  v.append (CrSim_im->dim ());
+	  
+	  v.append (CrSrAker_im_im->dim ());
+	  v.append (TrArSker_im_im->dim ());
+	  
+	  v.append (Aker_cap_Sker->dim ());
+	  
+	  v.append (Aim_cap_Tker->dim ());
+	  v.append (Bker_cap_Tker->dim ());
+	  v.append (ArSker_im_cap_Tker->dim ());
+	  
+	  v.append (Sim_cap_Bim->dim ());
+	  v.append (Sim_cap_BrTker_im->dim ());
+	  v.append (Sim_cap_Cker->dim ());
+	  v.append (SrAker_im_cap_Bim->dim ());
+	  v.append (SrAker_im_cap_BrTker_im->dim ());
+	  v.append (SrAker_im_cap_Cker->dim ());
+	  
+	  v.append (Tim_cap_Cim->dim ());
+	  v.append (TrAim_im_cap_Cim->dim ());
+	  v.append (TrBker_im_cap_Cim->dim ());
+	  v.append (TrArSker_im_im_cap_Cim->dim ());
+	  
+	  v.append (Tim_cap_CrSim_im->dim ());
+	  v.append (TrAim_im_cap_CrSim_im->dim ());
+	  v.append (TrBker_im_cap_CrSim_im->dim ());
+	  v.append (TrArSker_im_im_cap_CrSim_im->dim ());
+
+	  v.append (Tim_cap_CrSim_im->dim ());
+	  v.append (TrAim_im_cap_CrSim_im->dim ());
+	  v.append (TrBker_im_cap_CrSim_im->dim ());
+	  v.append (TrArSker_im_im_cap_CrSim_im->dim ());
+	  
+	  st.push (hq, v);
+	}
+      newline ();
+      
+      multivariate_laurentpoly<Z> P = H->free_poincare_polynomial ();
+      pair<pair<knot_desc,
+		map<grading, basedvector<int, 1> > > &,
+	   bool> p = P_sq1_knot_st.find (pair<multivariate_laurentpoly<Z>,
+					      map<grading, unsigned> > (P, sq1_ranks));
+      if (p.second)
+	{
+	  collisons[hwidth] ++;
+	  Ps += P;
+	  
+	  if (p.first.second != st)
+	    {
+	      printf ("DIFFER:\n");
+	      printf ("hwidth = %d\n", hwidth);
+	      
+	      show_st (knot_kh_sq, p.first.first);
+	      show_st (knot_kh_sq, i.key ());
+
+	      printf ("Kh[");
+	      planar_diagram (p.first.first.diagram ()).show_knottheory ();
+	      printf (", Modulus -> Null][q,t] === Kh[");
+	      planar_diagram (i.key ().diagram ()).show_knottheory ();
+	      printf (", Modulus -> Null][q,t]\n");
+	      
+#if 0
+	      printf ("%s:\n", 
+		      p.first.first.name ().c_str ());
+	      for (map<grading, basedvector<int, 1> >::const_iter j = p.first.second; j; j ++)
+		{
+		  printf ("  (%d, %d) -> [",
+			  j.key ().h, j.key ().q);
+		  for (unsigned k = 1; k <= j.val ().size (); k ++)
+		    {
+		      if (k > 1)
+			printf (",");
+		      printf ("%d", j.val ()[k]);
+		    }
+		  newline ();
+		}
+	      printf ("%s:\n",
+		      i.key ().name ().c_str ());
+	      for (map<grading, basedvector<int, 1> >::const_iter j = st; j; j ++)
+		{
+		  printf ("  (%d, %d) -> [",
+			  j.key ().h, j.key ().q);
+		  for (unsigned k = 1; k <= j.val ().size (); k ++)
+		    {
+		      if (k > 1)
+			printf (",");
+		      printf ("%d", j.val ()[k]);
+		    }
+		  newline ();
+		}
+#endif
+	    }
+	}
+      else
+	{
+	  p.first.first = i.key ();
+	  p.first.second = st;
+	}
+    }
+  
+  printf ("groups = %d\n", Ps.card ());
+  printf ("collisons:\n");
+  for (unsigned i = 1; i <= 10; i ++)
+    printf ("  % 2d: %d\n", i, collisons[i]);
+  
+#endif
+#endif
+
+#if 0
+#if 0
+  reader r ("sqtest.dat");
+  
+  knot_desc kd (knot_desc::ROLFSEN, 8, 19);
+  mod_map<Z2> sq1 (r);
+  mod_map<Z2> sq2 (r);
+  
+  ptr<const module<Z2> > H = sq1.domain ();
+  
+  display ("sq1:\n", sq1);
+  display ("sq2:\n", sq2);
+#endif
+  
+  for (unsigned i = 1; i <= 10; i ++)
+    for (unsigned j = 1; j <= rolfsen_crossing_knots (i);  j ++)
+      {
+#if 0
+	// (Knot Atlas) L10n18 
+	int xings[10][4] = {
+	  { 6, 1, 7, 2 },
+	  { 18, 7, 19, 8 },
+	  { 4,19,1,20 },
+	  { 5,12,6,13 },
+	  { 8,4,9,3 },
+	  { 13,17,14,16 },
+	  { 9,15,10,14 },
+	  { 15,11,16,10 },
+	  { 11,20,12,5 },
+	  { 2,18,3,17 },
+	};
+	knot_diagram kd (planar_diagram ("L10n18", 10, xings));
+#endif
+	
+	// (Knot Atlas) L10n102
+	int xings[10][4] = {
+	  { 6,1,7,2 }, { 10,3,11,4 }, { 14,7,15,8 }, { 8,13,5,14 }, { 11,18,12,19 }, { 15,20,16,17 }, { 19,16,20,9 }, { 17,12,18,13 }, { 2,5,3,6 }, { 4,9,1,10 },
+	};
+	knot_diagram kd (planar_diagram ("L10n102", 10, xings));
+	
+	// knot_diagram kd (rolfsen_knot (i, j));
+	// knot_diagram kd (mt_link (10, 0, 18));
+	
+	// show (kd); newline ();
+	printf ("%s  ", kd.name.c_str ());
+	
+	cube<Z2> c (kd);
+	mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
+	
+	chain_complex_simplifier<Z2> s (c.khC, d, 1);
+	
+	steenrod_square sq (c, d, s);
+
+	mod_map<Z2> sq1 = sq.sq1 ();
+	mod_map<Z2> sq2 = sq.sq2 ();
+	
+	assert (sq1.compose (sq1) == 0);
+	assert (sq2.compose (sq2) + sq1.compose (sq2).compose (sq1) == 0);
+	
+	ptr<const module<Z2> > H = sq1.domain ();
+	
+	bool first = 1;
+  
+	// ???
+	set<grading> gs = H->gradings ();
+	for (set_const_iter<grading> i = gs; i; i ++)
+	  {
+	    grading hq = i.val (),
+	      h1q (hq.h + 1, hq.q),
+	      h2q (hq.h + 2, hq.q);
+      
+	    // printf ("(%d, %d):\n", hq.h, hq.q);
+      
+	    ptr<const free_submodule<Z2> > H_hq = H->graded_piece (hq),
+	      H_h1q = H->graded_piece (h1q),
+	      H_h2q = H->graded_piece (h2q);
+      
+	    mod_map<Z2> whole = sq2.restrict (H_hq, H_h2q),
+	      tail = sq1.restrict (H_hq, H_h1q),
+	      head = sq1.restrict (H_h1q, H_h2q);
+      
+	    ptr<const free_submodule<Z2> > whole_im = whole.image (),
+	      tail_ker = tail.kernel (),
+	      head_im = head.image ();
+	    ptr<const free_submodule<Z2> > inter = whole_im->intersection (head_im);
+      
+	    mod_map<Z2> whole_res = whole.restrict_from (tail_ker);
+	    ptr<const free_submodule<Z2> > whole_res_im = whole_res.image ();
+      
+	    ptr<const free_submodule<Z2> > res_inter = whole_res_im->intersection (head_im);
+      
+	    int r1 = whole_im->dim ();
+	    int r2 = whole_res_im->dim ();
+	    int r3 = inter->dim ();
+	    int r4 = res_inter->dim ();
+      
+	    if (r1 == 0
+		&& r2 == 0
+		&& r3 == 0
+		&& r4 == 0)
+	      continue;
+      
+	    // printf ("  r = (%d, %d, %d, %d)\n", r1, r2, r3, r4);
+      
+	    int s1 = r2 - r4,
+	      s2 = r1 - r2 - r3 + r4,
+	      s3 = r4,
+	      s4 = r3 - r4;
+
+	    if (first)
+	      {
+#if 0
+		show (kd);
+		printf (":  ");
+#endif
+		first = 0;
+	      }
+	    else
+	      printf (", ");
+	    printf ("(%d, %d) -> (%d, %d, %d, %d)",
+		    hq.h, hq.q,
+		    s1, s2, s3, s4);
+	  }
+	// if (!first)
+	  newline ();
+      }
+#endif
+
+#if 0
+  // knot_diagram kd (htw_knot (12, 0, 48));
+  knot_diagram kd (htw_knot (10, 1, 23));
+  show (kd); newline ();
+#endif
+  
+#if 0
+  for (int a = 1; a >= 0; a --)
+    for (unsigned i = 1; i <= 9; i ++)
+      for (unsigned j = 1; j <= htw_knots (i, a); j ++)
+	{
+	  knot_diagram kd (htw_knot (i, a, j));
+	  show (kd); newline ();
+	  
+	  cube<Z2> c (kd);
+	  mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
+  
+	  chain_complex_simplifier<Z2> s (c.khC, d, 1);
+  
+	  steenrod_square sq (c, d, s);
+	  mod_map<Z2> sq1 = sq.sq1 ();
+	  // display ("sq1:\n", sq1);
+
+	  mod_map<Z2> sq2 = sq.sq2 ();
+	  if (a)
+	    assert (sq2 == 0);
+	  else
+	    display ("sq2:\n", sq2);
+  
+	  assert (sq1.compose (sq1) == 0);
+	  assert (sq2.compose (sq2) + sq1.compose (sq2).compose (sq1) == 0);
+	}
+#endif
+  
+#if 0
+  typedef Z2 R;
+  
+  for (unsigned i = 12; i <= 12; i ++)
     for (unsigned j = 1; j <= htw_knots (i, 0); j ++)
       {
 	knot_diagram kd (htw_knot (i, 0, j));
+	show (kd); newline ();
 	
+	cube<R> c (kd);
+	mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
+	
+	chain_complex_simplifier<R> s (c.khC, d, 1);
+	// assert (s.new_d == 0);
+	
+	printf ("|s.new_C| = %d\n", s.new_C->dim ());
+      }
+#endif
+  
+#if 0
+  map<multivariate_laurentpoly<Z>,
+      set<triple<unsigned, int, unsigned> > > kh_knot_map;
+  
+  for (int a = 1; a >= 0; a --)
+    for (unsigned i = 1; i <= 12; i ++)
+      for (unsigned j = 1; j <= htw_knots (i, a); j ++)
+	{
+	  knot_diagram kd (htw_knot (i, a, j));
+	  kd.marked_edge = 1;
+	  show (kd); newline ();
+	  
+	  cube<Z2> c (kd, 1);
+	  mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
+	  sseq_builder b (c.khC, d);
+	  sseq ss = b.build_sseq ();
+	  
+	  multivariate_laurentpoly<Z> P = ss.pages[1].poincare_polynomial (ss.bounds);
+	  kh_knot_map[P].push (triple<unsigned, int, unsigned> (i, a, j));
+	}
+  
+  {
+    writer w ("kh_knot_map.dat");
+    write (w, kh_knot_map);
+  }
+#endif
+  
+#if 0
+  compute_show_kh_sq (knot_desc (knot_desc::ROLFSEN, 8, 19));
+  
+  map<knot_desc,
+      triple<multivariate_laurentpoly<Z>,
+	     multivariate_laurentpoly<Z>,
+	     multivariate_laurentpoly<Z> > > knot_kh_sq;
+  
+  for (unsigned i = 1; i <= 10; i ++)
+    for (unsigned j = 1; j <= rolfsen_crossing_knots (i); j += 4000)
+      {
+	load (knot_kh_sq, knot_desc (knot_desc::ROLFSEN, i, j));
+      }
+  
+  for (unsigned i = 1; i <= 12; i ++)
+    for (unsigned j = 1; j <= htw_knots (i); j += 4000)
+      {
+	load (knot_kh_sq, knot_desc (knot_desc::HTW, i, j));
+      }
+  
+  for (unsigned i = 1; i <= 13; i ++)
+    for (unsigned j = 1; j <= mt_links (i); j += 4000)
+      {
+	load (knot_kh_sq, knot_desc (knot_desc::MT, i, j));
+      }
+  
+  map<multivariate_laurentpoly<Z>,
+      triple<knot_desc,
+	     multivariate_laurentpoly<Z>,
+	     multivariate_laurentpoly<Z> > > m;
+  
+  multivariate_laurentpoly<Z> groups;
+  
+  for (map<knot_desc,
+	   triple<multivariate_laurentpoly<Z>,
+		  multivariate_laurentpoly<Z>,
+		  multivariate_laurentpoly<Z> > >::const_iter i = knot_kh_sq; i; i ++)
+    {
+      multivariate_laurentpoly<Z> P = i.val ().first,
+	P_sq1 = i.val ().second,
+	P_sq2 = i.val ().third;
+      
+      pair<triple<knot_desc,
+		  multivariate_laurentpoly<Z>,
+		  multivariate_laurentpoly<Z> > &,
+	   bool> p = m.find (P);
+      if (p.second)
+	{
+	  groups += P;
+	  
+	  if (p.first.second != P_sq1
+	      || p.first.third != P_sq2)
+	    {
+	      printf ("DIFFER  ");
+	      display (P);
+	      
+	      printf ("  %s\n", p.first.first.name ().c_str ());
+	      printf ("    sq1 "); display (p.first.second);
+	      printf ("    sq2 "); display (p.first.third);
+
+	      printf ("  %s\n", i.key ().name ().c_str ());
+	      printf ("    sq1 "); display (P_sq1);
+	      printf ("    sq2 "); display (P_sq2);
+	    }
+	}
+      else
+	{
+	  p.first.first = i.key ();
+	  p.first.second = P_sq1;
+	  p.first.third = P_sq2;
+	}
+    }
+  
+  printf ("|groups| = %d\n", groups.card ());
+  
+  printf ("done.\n");
+#endif
+  
+#if 0
+  reader r ("kh_knot_map.dat");
+  map<multivariate_laurentpoly<Z>,
+      set<triple<unsigned, int, unsigned> > > kh_knot_map (r);
+  
+  for (map<multivariate_laurentpoly<Z>,
+	   set<triple<unsigned, int, unsigned> > >::const_iter i = kh_knot_map; i; i ++)
+    {
+      if (i.val ().card () == 1)
+	continue;
+      
+      printf ("group\n");
+      bool first = 1;
+      multivariate_laurentpoly<Z> P, sq1_P, sq2_P;
+      
+      for (set_const_iter<triple<unsigned, int, unsigned> > j = i.val (); j; j ++)
+	{
+	  knot_diagram kd (htw_knot (j.val ().first,
+				     j.val ().second,
+				     j.val ().third));
+	  printf ("  "); show (kd); newline ();
+	  
+	  triple<multivariate_laurentpoly<Z>,
+		 multivariate_laurentpoly<Z>,
+		 multivariate_laurentpoly<Z> > t = square (kd);
+#if 0
+	  display ("t.first = ", t.first);
+	  display ("i.key () = ", i.key ());
+	  assert (t.first == i.key ());
+#endif
+	  
+	  if (first)
+	    {
+	      P = t.first;
+	      sq1_P = t.second;
+	      sq2_P = t.third;
+	      first = 0;
+	    }
+	  else
+	    {
+	      assert (P == t.first);
+	      if (sq1_P != t.second)
+		printf ("  prev sq1_P != sq1_P\n");
+	      if (sq2_P != t.third)
+		printf ("  prev sq2_P != sq2_P\n");
+	    }
+	}
+    }
+#endif
+  
+#if 0
+  typedef Z2 F;
+  typedef fraction_field<polynomial<F> > R;
+
+  for (unsigned i = 1; i <= 10; i ++)
+    for (unsigned j = 1; j <= htw_knots (i, 0); j ++)
+      {
+	knot_diagram kd (htw_knot (i, 0, j));
 	kd.marked_edge = 1;
 	
 	show (kd); newline ();
 	
-	spanning_tree_complex<Z2> c (kd);
+	spanning_tree_complex<F> st (kd);
   
-	mod_map<fraction_field<polynomial<Z2> > > E2_d = c.twisted_d2 ();
-	assert (E2_d.compose (E2_d) == 0);
-	// display ("E2_d:\n", E2_d);
+	mod_map<R> d2 = st.twisted_d2 ();
+	assert (d2.compose (d2) == 0);
 	
-	chain_complex_simplifier<fraction_field<polynomial<Z2> > > E2_s (c.C, E2_d, 2);
-	assert (E2_s.new_d == 0);
+	mod_map<R> d2U1 = st.twisted_d2Un (1);
+	// mod_map<R> d2U1 = st.twisted_d2U1_test ();
   
-	multivariate_laurentpoly<Z> E3_p = E2_s.new_C->free_delta_poincare_polynomial ();
-	printf ("E3_p = "); show (E3_p); newline ();
-  
-	mod_map<fraction_field<polynomial<Z2> > > tt_d = c.totally_twisted_kh_d ();
-	assert (tt_d.compose (tt_d) == 0);
-	// display ("tt_d:\n", tt_d);
+	assert (d2.compose (d2U1) + d2U1.compose (d2) == 0);
 	
-	chain_complex_simplifier<fraction_field<polynomial<Z2> > > tt_s (c.C, tt_d, 2);
-	assert (tt_s.new_d == 0);
+	mod_map<R> d2U2 = st.twisted_d2Un (2);
+	assert (d2.compose (d2U2) + d2U2.compose (d2) + d2U1.compose (d2U1) == 0);
+  
+	mod_map<R> d2U3 = st.twisted_d2Un (3);
+	assert (d2.compose (d2U3) + d2U3.compose (d2)
+		+ d2U2.compose (d2U1) + d2U1.compose (d2U2) == 0);
 
-	multivariate_laurentpoly<Z> tt_p = tt_s.new_C->free_delta_poincare_polynomial ();
-	printf ("tt_p = "); show (tt_p); newline ();
+	mod_map<R> d2U4 = st.twisted_d2Un (4);
+	assert (d2.compose (d2U4) + d2U4.compose (d2)
+		+ d2U3.compose (d2U1) + d2U1.compose (d2U3)
+		+ d2U2.compose (d2U2) == 0);
+
+	mod_map<R> d2U5 = st.twisted_d2Un (5);
+	assert (d2.compose (d2U5) + d2U5.compose (d2)
+		+ d2U4.compose (d2U1) + d2U1.compose (d2U4)
+		+ d2U3.compose (d2U2) + d2U2.compose (d2U3) == 0);
 	
-	cube<Z2> kh_c (kd, 1);
-	mod_map<Z2> kh_d = kh_c.compute_d (1, 0, 0, 0, 0);
-	
-	sseq_builder builder (kh_c.khC, kh_d);
-	sseq ss = builder.build_sseq ();
-	multivariate_laurentpoly<Z> kh_p = ss.pages[1].delta_poincare_polynomial (ss.bounds);
-	printf ("kh_p = "); show (kh_p); newline ();
-	
-	if (tt_p != kh_p)
-	  printf (" > tt_p != kh_p!!\n");
-	
-	if (! rank_lte (E3_p, tt_p))
-	  printf (" > rank E2 > rank tt!!\n");
+	mod_map<R> d2U6 = st.twisted_d2Un (6);
+	assert (d2.compose (d2U6) + d2U6.compose (d2)
+		+ d2U5.compose (d2U1) + d2U1.compose (d2U5)
+		+ d2U4.compose (d2U2) + d2U2.compose (d2U4)
+		+ d2U3.compose (d2U3) == 0);
       }
+#endif
 }
diff --git a/mpi_aux.cpp b/mpi_aux.cpp
new file mode 100644
index 0000000..724f4b5
--- /dev/null
+++ b/mpi_aux.cpp
@@ -0,0 +1,133 @@
+
+#include <mpi.h>
+#include <mpi_aux.h>
+
+// #include <knotkit.h>
+
+#define DATATAG 1
+
+void
+comm_init (int *argc, char ***argv)
+{
+  MPI_Init (argc, argv);
+}
+
+void
+comm_finalize ()
+{
+  MPI_Finalize ();
+}
+
+int
+self_rank ()
+{
+  int node;
+  MPI_Comm_rank (MPI_COMM_WORLD, &node);
+  return node;
+}
+
+int
+num_tasks  ()
+{
+  int ntasks;
+  MPI_Comm_size (MPI_COMM_WORLD, &ntasks);
+  return ntasks;
+}
+
+void
+send_int (int v, int dest)
+{
+  MPI_Send (&v, 1, MPI_INT, dest, DATATAG, MPI_COMM_WORLD);
+}
+
+void
+send_string (const char *s, int dest)
+{
+  int n = strlen (s);
+  MPI_Send (&n, 1, MPI_INT, dest, DATATAG, MPI_COMM_WORLD);
+  MPI_Send ((void *)s, n, MPI_CHAR, dest, DATATAG, MPI_COMM_WORLD);
+}
+
+void
+send_string (const std::string &s, int dest)
+{
+  send_string (s.c_str (), dest);
+}
+
+int
+recv_int (int *src)
+{
+  int v;
+  MPI_Status status;
+  MPI_Recv (&v,                /* message buffer */
+	    1,                 /* one data item */
+	    MPI_INT,           /* of type int */
+	    MPI_ANY_SOURCE,    /* receive from any sender */
+	    MPI_ANY_TAG,       /* any type of message */
+	    MPI_COMM_WORLD,    /* default communicator */
+	    &status);          /* info about the received message */
+  if (src)
+    *src = status.MPI_SOURCE;
+  return v;
+}
+
+std::string
+recv_string (int *src)
+{
+  int n;
+  MPI_Status status;
+  MPI_Recv (&n,                /* message buffer */
+	    1,                 /* one data item */
+	    MPI_INT,           /* of type int */
+	    MPI_ANY_SOURCE,    /* receive from any sender */
+	    MPI_ANY_TAG,       /* any type of message */
+	    MPI_COMM_WORLD,    /* default communicator */
+	    &status);          /* info about the received message */
+  char *s = new char[n + 1];
+  MPI_Recv (s,                 /* message buffer */
+	    n,                 /* one data item */
+	    MPI_CHAR,          /* of type int */
+	    status.MPI_SOURCE, /* receive from any sender */
+	    MPI_ANY_TAG,       /* any type of message */
+	    MPI_COMM_WORLD,    /* default communicator */
+	    &status);          /* info about the received message */
+  if (src)
+    *src = status.MPI_SOURCE;
+  s[n] = '\0';
+  std::string s_std (s);
+  delete [] s;
+  return s_std;
+}
+
+#if 0
+void
+send_htw_knot (unsigned n, bool alternating, unsigned k, int dest)
+{
+  int data[3];
+  data[0] = (int)n;
+  data[1] = (int)alternating;
+  data[2] = (int)k;
+  MPI_Send (&data, 3, MPI_INT, dest, DATATAG, MPI_COMM_WORLD);
+}
+
+knot_diagram
+recv_knot (int *src)
+{
+  int data[3];
+  MPI_Status status;
+  MPI_Recv (&data,             /* message buffer */
+	    3,                 /* one data item */
+	    MPI_INT,           /* of type int */
+	    MPI_ANY_SOURCE,    /* receive from any sender */
+	    MPI_ANY_TAG,       /* any type of message */
+	    MPI_COMM_WORLD,    /* default communicator */
+	    &status);          /* info about the received message */
+  if (src)
+    *src = status.MPI_SOURCE;
+  
+  dt_code k = htw_knot ((unsigned)data[0],
+			(bool)data[1],
+			(unsigned)data[2]);
+  return knot_diagram (k);
+}
+#endif
diff --git a/mpi_aux.h b/mpi_aux.h
new file mode 100644
index 0000000..ed80442
--- /dev/null
+++ b/mpi_aux.h
@@ -0,0 +1,22 @@
+
+void comm_init (int *argc, char ***argv);
+void comm_finalize ();
+
+int self_rank ();
+int num_tasks  ();
+
+void send_int (int v, int dest);
+void send_string (const char *s, int dest);
+void send_string (const std::string &s, int dest);
+
+int recv_int (int *src);
+std::string recv_string (int *src);
+
+inline int recv_int () { return recv_int (0); }
+inline std::string recv_string () { return recv_string (0); }
+
+#if 0
+void send_htw_knot (unsigned n, bool alternating, unsigned k, int dest);
+knot_diagram recv_knot (int *src);
+knot_diagram recv_knot () { return recv_knot (0); }
+#endif
diff --git a/mpimain.cpp b/mpimain.cpp
new file mode 100644
index 0000000..81fefa8
--- /dev/null
+++ b/mpimain.cpp
@@ -0,0 +1,578 @@
+
+#include <knotkit.h>
+
+#include <mpi.h>
+#include <mpi_aux.h>
+
+#define CMD_DO 1
+#define CMD_DIE 2
+
+static const int block_size = 100;
+
+void
+master ()
+{
+  basedvector<knot_desc, 1> work;
+  
+  for (unsigned i = 1; i <= 10; i ++)
+    for (unsigned j = 1; j <= mt_links (i); j ++)
+      {
+	knot_diagram kd (mt_links (i, j));
+	unsigned n = kd.num_components ();
+	if (n < 2)
+	  continue;
+	
+	work.append (knot_desc (knot_desc::MT, i, j));
+      }
+  
+#if 0  
+  basedvector<basedvector<unsigned, 1>, 1> groups = mutant_knot_groups (15);
+
+  for (unsigned i = 1; i <= groups.size (); i ++)
+    for (unsigned j = 1; j <= groups[i].size (); j ++)
+      work.append (knot_desc (knot_desc::HTW, 15, groups[i][j]));
+#endif
+  
+#if 0
+  for (unsigned i = 14; i >= 1; i --)
+    {
+      if (i <= 10)
+	{
+	  for (unsigned j = 1; j <= rolfsen_crossing_knots (i); j += block_size)
+	    {
+	      work.append (knot_desc (ROLFSEN, i, j));
+	    }
+	}
+      
+      for (unsigned j = 1; j <= htw_knots (i); j += block_size)
+	{
+	  work.append (knot_desc (HTW, i, j));
+	}
+      
+      if (i <= 13)
+	{
+	  for (unsigned j = 1; j <= mt_links (i); j += block_size)
+	    {
+	      work.append (knot_desc (MT, i, j));
+	    }
+	}
+    }
+#endif
+  
+  int ntasks = num_tasks ();
+  
+  set<unsigned> active;
+  
+  for (int rank = 1; rank < ntasks && work.size () > 0; rank ++)
+    {
+      send_int (CMD_DO, rank);
+      
+      knot_desc desc = work.pop ();
+      send_int ((int)desc.t, rank);
+      send_int ((int)desc.i, rank);
+      send_int ((int)desc.j, rank);
+      
+      active.push (rank);
+    }
+  while (work.size () > 0)
+    {
+      int rank;
+      int dummy = recv_int (&rank);
+      
+      send_int (CMD_DO, rank);
+      
+      knot_desc desc = work.pop ();
+      send_int ((int)desc.t, rank);
+      send_int ((int)desc.i, rank);
+      send_int ((int)desc.j, rank);
+    }
+  
+  while (active.card () > 0)
+    {
+      int rank;
+      int dummy = recv_int (&rank);
+      active -= rank;
+    }
+  
+  for (int rank = 1; rank < ntasks; rank ++)
+    {
+      send_int (CMD_DIE, rank);
+    }
+}
+
+bool
+file_exists (const char *file)
+{
+  struct stat stat_buf;
+  if (stat (file, &stat_buf) != 0)
+    {
+      if (errno == ENOENT)
+	return 0;
+      
+      stderror ("stat: %s", file);
+      exit (EXIT_FAILURE);
+    }
+  
+  return 1;
+}
+
+#if 0
+void
+compute_forgetful (int rank, knot_desc desc, const char *buf)
+{
+  knot_diagram kd = desc.diagram ();
+  
+  typedef Zp<97> R;
+  
+  unsigned n = kd.num_components ();
+  assert (n >= 2);
+  assert (n < 97);
+  
+  printf ("[% 2d] ", rank);
+  show (kd);
+  newline ();
+  
+  unionfind<1> u (kd.num_edges ());
+  for (unsigned i = 1; i <= kd.n_crossings; i ++)
+    {
+      u.join (kd.ept_edge (kd.crossings[i][1]),
+	      kd.ept_edge (kd.crossings[i][3]));
+      u.join (kd.ept_edge (kd.crossings[i][2]),
+	      kd.ept_edge (kd.crossings[i][4]));
+    }
+	
+  map<unsigned, unsigned> root_comp;
+  unsigned t = 0;
+  for (unsigned i = 1; i <= kd.num_edges (); i ++)
+    {
+      if (u.find (i) == i)
+	{
+	  ++ t;
+	  root_comp.push (i, t);
+	}
+    }
+  assert (t == n);
+	
+  basedvector<R, 1> comp_weight (n);
+  for (unsigned i = 1; i <= n; i ++)
+    comp_weight[i] = R ((int)i);
+	
+  map<unsigned, R> crossing_over_sign;
+	
+  // crossings
+  set<unsigned> pending;
+  set<unsigned> finished;
+	
+  crossing_over_sign.push (1, R (1));
+  pending.push (1);
+  
+  while (pending.card () > 0)
+    {
+      unsigned x = pending.pop ();
+      finished.push (x);
+      
+      R s = crossing_over_sign(x);
+      
+      for (unsigned j = 1; j <= 4; j ++)
+	{
+	  unsigned p = kd.crossings[x][j];
+	  R t = kd.is_over_ept (p) ? s : -s;  // sign of (x, p)
+	  
+	  unsigned q = kd.edge_other_ept (p);
+	  unsigned x2 = kd.ept_crossing[q];
+	  
+	  R u = kd.is_over_ept (q) ? -t : t;
+	  
+	  if (crossing_over_sign % x2)
+	    assert (crossing_over_sign(x2) == u);
+	  else
+	    crossing_over_sign.push (x2, u);
+	  
+	  if (! (finished % x2))
+	    pending += x2;
+	}
+    }
+  assert (finished.card () == kd.n_crossings);
+	
+#if 0
+  printf ("crossing_over_sign:\n");
+  for (unsigned i = 1; i <= kd.n_crossings; i ++)
+    {
+      printf ("  % 2d: ", i);
+      display (crossing_over_sign(i));
+    }
+#endif
+	
+  cube<R> c (kd);
+#if 0
+  for (unsigned i = 0; i < c.n_resolutions; i ++)
+    {
+      smallbitset state (kd.n_crossings, i);
+      smoothing s (kd, state);
+      s.show_self (kd, state);
+      newline ();
+    }
+#endif
+  
+  mod_map<R> untwisted_d = c.compute_d (1, 0, 0, 0, 0);
+  assert (untwisted_d.compose (untwisted_d) == 0);
+	
+  mod_map<R> d = untwisted_d;
+  for (unsigned x = 1; x <= kd.n_crossings; x ++)
+    {
+      unsigned p1 = kd.crossings[x][1],
+	p2 = kd.crossings[x][2];
+      assert (kd.is_over_ept (p2));
+	    
+      unsigned r1 = u.find (kd.ept_edge (p1)),
+	r2 = u.find (kd.ept_edge (p2));
+	    
+      unsigned c1 = root_comp(r1),
+	c2 = root_comp(r2);
+	    
+      if (c1 != c2)
+	{
+	  R s = crossing_over_sign(x);
+		
+	  R w_under = comp_weight[c1];
+	  R w_over = comp_weight[c2];
+		
+	  d = d + c.compute_dinv (x)*(s*(w_over - w_under))
+	    ;
+	}
+    }
+  
+  // display ("d:\n", d);
+  
+  // mod_map<Z> h = d.compose (d);
+  // display ("h:\n", h);
+  
+  assert (d.compose (d) == 0);
+  
+  ptr<const module<R> > Ek = c.khC;
+  mod_map<R> dk = d;
+  
+  basedvector<multivariate_laurentpoly<Z>, 1> pages;
+  
+  unsigned kinf;
+  for (int dq = 0;; dq -= 2)
+    {
+      chain_complex_simplifier<R> s (Ek, dk, dq);
+      Ek = s.new_C;
+      dk = s.new_d;
+      
+      pages.append (Ek->free_poincare_polynomial ());
+      
+      printf ("[% 2d] ", rank);
+      printf ("|E%d| = %d\n", ((- dq) / 2) + 1, Ek->dim ());
+      if (dk == 0)
+	{
+	  kinf = ((- dq) / 2);
+	  break;
+	}
+    }
+  
+  unsigned total_lk = kd.total_linking_number ();
+  
+  unsigned trivial_bound = total_lk == 0 ? 2 : total_lk;
+  
+  printf ("[% 2d] ", rank);
+  printf ("kinf = %d, trivial bound = %d (total_lk = %d)\n",
+	  kinf, trivial_bound, total_lk);
+  
+  if (trivial_bound < kinf)
+    {
+      printf ("[% 2d] ", rank);
+      printf (" > BETTER\n");
+    }
+  fflush (stdout);
+  
+  assert (desc.t == knot_desc::MT);  
+  gzfile_writer w (buf);
+  write (w, desc);
+  write (w, pages);
+}
+#endif
+
+unsigned
+compute_b_lk_weak (knot_diagram &kd)
+{
+  unsigned m = kd.num_components ();
+  assert (m > 1);
+  
+  if (m == 2)
+    {
+      unsigned total_lk = kd.total_linking_number ();
+      return total_lk == 0 ? 2 : total_lk;
+    }
+  
+  unionfind<1> u (kd.num_edges ());
+  
+  for (unsigned i = 1; i <= kd.n_crossings; i ++)
+    {
+      u.join (kd.ept_edge (kd.crossings[i][1]),
+	      kd.ept_edge (kd.crossings[i][3]));
+      u.join (kd.ept_edge (kd.crossings[i][2]),
+	      kd.ept_edge (kd.crossings[i][4]));
+    }
+  assert (m == u.num_sets ());
+  
+  map<unsigned, unsigned> root_comp;
+  unsigned t = 0;
+  for (unsigned i = 1; i <= kd.num_edges (); i ++)
+    {
+      if (u.find (i) == i)
+	{
+	  ++ t;
+	  root_comp.push (i, t);
+	}
+    }
+  assert (t == m);
+  
+  unsigned b_lk_weak = 0;
+  
+  for (unsigned i = 1; i <= m; i ++)
+    for (unsigned j = i + 1; j <= m; j ++)
+      {
+	assert (i < j);
+	
+	int lk = 0;
+	for (unsigned x = 1; x <= kd.n_crossings; x ++)
+	  {
+	    unsigned r1 = root_comp(u.find (kd.ept_edge (kd.crossings[x][1]))),
+	      r2 = root_comp(u.find (kd.ept_edge (kd.crossings[x][2])));
+	    if (((r1 == i) && (r2 == j))
+		|| ((r2 == i) && (r1 == j)))
+	      {
+		if (kd.is_to_ept (kd.crossings[x][1]) == kd.is_to_ept (kd.crossings[x][4]))
+		  lk ++;
+		else
+		  lk --;
+	      }
+	  }
+	assert (is_even (lk));
+	lk /= 2;
+	
+	if (lk == 0)
+	  {
+	    smallbitset ci (m);
+	    ci.push (i);
+	    
+	    smallbitset cj (m);
+	    cj.push (j);
+	    
+	    smallbitset c (m);
+	    c.push (i);
+	    c.push (j);
+	    knot_diagram Lij (SUBLINK, c, kd);
+	    
+	    multivariate_laurentpoly<Z> P_Lij = Kh_poincare_polynomial<Z2> (Lij);
+	    
+	    knot_diagram Li_join_Lj (DISJOINT_UNION,
+				     knot_diagram (SUBLINK, ci, kd),
+				     knot_diagram (SUBLINK, cj, kd));
+	    
+	    multivariate_laurentpoly<Z> P_Li_join_Lj = Kh_poincare_polynomial<Z2> (Li_join_Lj);
+	    
+	    if (P_Lij != P_Li_join_Lj)
+	      lk = 2;  // non-split
+	  }
+	
+	b_lk_weak += abs (lk);
+      }
+  
+  return b_lk_weak == 0 ? 2 : b_lk_weak;
+}
+
+void
+compute_splitting_bounds (const knot_desc &desc)
+{
+  typedef fraction_field<polynomial<Z2> > Z2x;
+  
+  int rank = self_rank ();
+  
+  knot_diagram kd = desc.diagram ();
+  unsigned m = kd.num_components ();
+  assert (m > 1);
+  
+  printf ("[% 2d] ", rank); show (kd); newline ();
+  printf ("[% 2d]  m = %d\n", rank, m);
+  
+  unsigned total_lk = kd.total_linking_number ();
+  printf ("[% 2d] total_lk = %d\n", rank, total_lk);
+  
+  unionfind<1> u (kd.num_edges ());
+  
+  for (unsigned i = 1; i <= kd.n_crossings; i ++)
+    {
+      u.join (kd.ept_edge (kd.crossings[i][1]),
+	      kd.ept_edge (kd.crossings[i][3]));
+      u.join (kd.ept_edge (kd.crossings[i][2]),
+	      kd.ept_edge (kd.crossings[i][4]));
+    }
+  assert (u.num_sets () == m);
+  
+  map<unsigned, unsigned> root_comp;
+  unsigned t = 0;
+  for (unsigned i = 1; i <= kd.num_edges (); i ++)
+    {
+      if (u.find (i) == i)
+	{
+	  ++ t;
+	  root_comp.push (i, t);
+	}
+    }
+  assert (t == m);
+  
+  basedvector<Q, 1> comp_weightQ (m);
+  for (unsigned i = 1; i <= m; i ++)
+    comp_weightQ[i] = Q (i);
+  unsigned bQ = splitting_bound<Q> (kd, comp_weightQ);
+  
+  basedvector<Z2x, 1> comp_weightZ2x (m);
+  for (unsigned i = 1; i <= m; i ++)
+    comp_weightZ2x[i] = Z2x (polynomial<Z2> (Z2 (1), i));
+  unsigned bZ2x = splitting_bound<Z2x> (kd, comp_weightZ2x);
+  
+  // lower bound
+  unsigned b = std::max (bQ, bZ2x);
+  
+  // adjust parity
+  if ((total_lk & 1) != (b & 1))
+    b ++;
+  
+  printf ("[% 2d] bQ = %d\n", rank, bQ);
+  printf ("[% 2d] bZ2x = %d\n", rank, bZ2x);
+  printf ("[% 2d] b = %d\n", rank, b);
+  
+  unsigned b_lk_weaker = total_lk == 0 ? 2 : total_lk; // kd is non-split
+  
+  unsigned b_lk_weak = compute_b_lk_weak (kd);
+  assert (b_lk_weaker <= b_lk_weak);
+  
+  printf ("[% 2d] b_lk_weaker = %d\n", rank, b_lk_weaker);
+  printf ("[% 2d] b_lk_weak = %d\n", rank, b_lk_weak);
+  
+  if (b_lk_weaker < b_lk_weak)
+    printf ("[% 2d] > STRICTLY WEAKER\n", rank);
+  
+  basedvector<basedvector<unsigned, 1>, 1> ps = permutations (m);
+  unsigned upper = kd.n_crossings;
+  for (unsigned i = 1; i <= ps.size (); i ++)
+    {
+      basedvector<unsigned, 1> p = ps[i];
+	    
+      unsigned ri = 0;
+      for (unsigned j = 1; j <= kd.n_crossings; j ++)
+	{
+	  unsigned upper_e = kd.ept_edge (kd.crossings[j][2]),
+	    lower_e = kd.ept_edge (kd.crossings[j][1]);
+		
+	  unsigned upper_c = root_comp(u.find (upper_e)),
+	    lower_c = root_comp(u.find (lower_e));
+		
+	  if (upper_c != lower_c
+	      && p[upper_c] < p[lower_c])
+	    ri ++;
+	}
+	    
+      if (ri < upper)
+	upper = ri;
+    }
+  printf ("[% 2d] upper = %d\n", rank, upper);
+  
+  assert (b_lk_weak <= upper);
+  assert (b <= upper);
+  
+  // non-trivial link, sp at least 1.
+  unsigned best = std::max (b, b_lk_weak);
+  
+  if (best == upper)
+    printf ("[% 2d] > sp = %d", rank, upper);
+  else
+    printf ("[% 2d] > %d <= sp <= %d", rank, best, upper);
+  
+  printf (" ");
+  
+  if (b == best
+      && b_lk_weak == best)
+    printf ("(b + b_lk_weak)");
+  else if (b == best)
+    printf ("(b)");
+  else
+    {
+      assert (b_lk_weak == best);
+      printf ("(b_lk_weak)");
+    }
+  fflush (stdout);
+  
+  char buf[1000];
+  assert (desc.t == knot_desc::MT);
+  sprintf (buf, "splitting/L%d_%d.dat.gz\n", desc.i, desc.j);
+  
+  {
+    gzfile_reader r (buf);
+    write (r, desc);
+    write (r, bQ);
+    write (r, bZ2x);
+    write (r, b);
+    write (r, b_lk_weak);
+    write (r, upper);
+  }
+}
+
+void
+slave ()
+{
+  int rank = self_rank ();
+  
+  for (;;)
+    {
+      int cmd = recv_int ();
+      switch (cmd)
+	{
+	case CMD_DO:
+	  {
+	    knot_desc desc;
+	    desc.t = (knot_desc::table)recv_int ();
+	    desc.i = (unsigned)recv_int ();
+	    desc.j = (unsigned)recv_int ();
+	    
+	    printf ("[% 2d] CMD_DO %s\n", rank, desc.name ().c_str ());
+	    
+	    compute_splitting_bounds (desc);
+	    
+	    send_int (0, 0);
+	  }
+	  break;
+	  
+	case CMD_DIE:
+	  return;
+	}
+    }
+}
+
+int
+main (int argc, char **argv)
+{
+  comm_init (&argc, &argv);
+  
+  int rank = self_rank (),
+    ntasks = num_tasks ();
+  
+  printf ("[% 2d] alive\n", rank);
+  fflush (stdout);
+  
+  if (rank == 0)
+    {
+      printf ("ntasks = %d\n", ntasks);
+      fflush (stdout);
+      
+      master ();
+    }
+  else
+    slave ();
+  
+  comm_finalize ();
+  return 0;
+}
diff --git a/parallel.cmd b/parallel.cmd
new file mode 100644
index 0000000..a074876
--- /dev/null
+++ b/parallel.cmd
@@ -0,0 +1,20 @@
+# mpi job, runtime max 5days
+#PBS -l nodes=3:ppn=4,walltime=150:00:00
+# xxx PBS -l mem=80gb
+#
+# sends mail if the process aborts, when it begins, and
+# when it ends (abe)
+#PBS -m abe
+#
+module load openmpi/gcc
+# go to the directory with the program
+cd $HOME/src/knotkit
+
+LD_LIBRARY_PATH=$LD_LIBRARY_PATH:/u/cseed/llvm-3.1/lib
+export LD_LIBRARY_PATH
+
+echo $LD_LIBRARY_PATH
+
+# and run it
+numprocs=`wc -l <${PBS_NODEFILE}`
+mpiexec -np $numprocs ./mpimain
diff --git a/planar_diagram.cpp b/planar_diagram.cpp
index 1a23554..650d667 100644
--- a/planar_diagram.cpp
+++ b/planar_diagram.cpp
@@ -19,21 +19,20 @@ planar_diagram::planar_diagram (const knot_diagram &kd)
 }
 
 void
-planar_diagram::display_bohua () const
+planar_diagram::show_knottheory () const
 {
-  printf ("%s\t[", name.c_str ());
+  printf ("PD[");
   for (unsigned i = 1; i <= crossings.size (); i ++)
     {
       if (i > 1)
 	printf (",");
-      printf ("[%d,%d,%d,%d]", 
+      printf ("X[%d,%d,%d,%d]", 
 	      crossings[i][1],
 	      crossings[i][2],
 	      crossings[i][3],
 	      crossings[i][4]);
     }
-  printf ("]\n");
-  
+  printf ("]");
 }
 
 void
diff --git a/planar_diagram.h b/planar_diagram.h
index cd095df..6c32c3a 100644
--- a/planar_diagram.h
+++ b/planar_diagram.h
@@ -21,8 +21,9 @@ public:
   { }
   ~planar_diagram () { }
   
-  void display_bohua () const;
-
+  void show_knottheory () const;
+  void display_knottheory () const { show_knottheory (); newline (); }
+  
   void show_self () const { printf ("planar_diagram %s", name.c_str ()); }
   void display_self () const;
 };
diff --git a/rd_parser/rd_scanner.cc b/rd_parser/rd_scanner.cc
index 372c166..6f19fc8 100644
--- a/rd_parser/rd_scanner.cc
+++ b/rd_parser/rd_scanner.cc
@@ -66,6 +66,7 @@ typedef int16_t flex_int16_t;
 typedef uint16_t flex_uint16_t;
 typedef int32_t flex_int32_t;
 typedef uint32_t flex_uint32_t;
+typedef uint64_t flex_uint64_t;
 #else
 typedef signed char flex_int8_t;
 typedef short int flex_int16_t;
@@ -172,7 +173,12 @@ typedef unsigned int flex_uint32_t;
 typedef struct yy_buffer_state *YY_BUFFER_STATE;
 #endif
 
-extern int rd_yyleng;
+#ifndef YY_TYPEDEF_YY_SIZE_T
+#define YY_TYPEDEF_YY_SIZE_T
+typedef size_t yy_size_t;
+#endif
+
+extern yy_size_t rd_yyleng;
 
 extern FILE *rd_yyin, *rd_yyout;
 
@@ -198,11 +204,6 @@ extern FILE *rd_yyin, *rd_yyout;
 
 #define unput(c) yyunput( c, (yytext_ptr)  )
 
-#ifndef YY_TYPEDEF_YY_SIZE_T
-#define YY_TYPEDEF_YY_SIZE_T
-typedef size_t yy_size_t;
-#endif
-
 #ifndef YY_STRUCT_YY_BUFFER_STATE
 #define YY_STRUCT_YY_BUFFER_STATE
 struct yy_buffer_state
@@ -220,7 +221,7 @@ struct yy_buffer_state
 	/* Number of characters read into yy_ch_buf, not including EOB
 	 * characters.
 	 */
-	int yy_n_chars;
+	yy_size_t yy_n_chars;
 
 	/* Whether we "own" the buffer - i.e., we know we created it,
 	 * and can realloc() it to grow it, and should free() it to
@@ -290,8 +291,8 @@ static YY_BUFFER_STATE * yy_buffer_stack = 0; /**< Stack as an array. */
 
 /* yy_hold_char holds the character lost when rd_yytext is formed. */
 static char yy_hold_char;
-static int yy_n_chars;		/* number of characters read into yy_ch_buf */
-int rd_yyleng;
+static yy_size_t yy_n_chars;		/* number of characters read into yy_ch_buf */
+yy_size_t rd_yyleng;
 
 /* Points to current character in buffer. */
 static char *yy_c_buf_p = (char *) 0;
@@ -319,7 +320,7 @@ static void rd_yy_init_buffer (YY_BUFFER_STATE b,FILE *file  );
 
 YY_BUFFER_STATE rd_yy_scan_buffer (char *base,yy_size_t size  );
 YY_BUFFER_STATE rd_yy_scan_string (yyconst char *yy_str  );
-YY_BUFFER_STATE rd_yy_scan_bytes (yyconst char *bytes,int len  );
+YY_BUFFER_STATE rd_yy_scan_bytes (yyconst char *bytes,yy_size_t len  );
 
 void *rd_yyalloc (yy_size_t  );
 void *rd_yyrealloc (void *,yy_size_t  );
@@ -377,7 +378,7 @@ static void yy_fatal_error (yyconst char msg[]  );
  */
 #define YY_DO_BEFORE_ACTION \
 	(yytext_ptr) = yy_bp; \
-	rd_yyleng = (size_t) (yy_cp - yy_bp); \
+	rd_yyleng = (yy_size_t) (yy_cp - yy_bp); \
 	(yy_hold_char) = *yy_cp; \
 	*yy_cp = '\0'; \
 	(yy_c_buf_p) = yy_cp;
@@ -483,7 +484,7 @@ char *rd_yytext;
 
 typedef yy::rd_parser::token token;
 typedef yy::rd_parser::token_type token_type;
-#line 487 "rd_parser/rd_scanner.cc"
+#line 488 "rd_parser/rd_scanner.cc"
 
 #define INITIAL 0
 
@@ -522,7 +523,7 @@ FILE *rd_yyget_out (void );
 
 void rd_yyset_out  (FILE * out_str  );
 
-int rd_yyget_leng (void );
+yy_size_t rd_yyget_leng (void );
 
 char *rd_yyget_text (void );
 
@@ -581,7 +582,7 @@ static int input (void );
 	if ( YY_CURRENT_BUFFER_LVALUE->yy_is_interactive ) \
 		{ \
 		int c = '*'; \
-		int n; \
+		yy_size_t n; \
 		for ( n = 0; n < max_size && \
 			     (c = getc( rd_yyin )) != EOF && c != '\n'; ++n ) \
 			buf[n] = (char) c; \
@@ -666,7 +667,7 @@ YY_DECL
 #line 16 "rd_parser/rd_scanner.ll"
 
 
-#line 670 "rd_parser/rd_scanner.cc"
+#line 671 "rd_parser/rd_scanner.cc"
 
 	if ( !(yy_init) )
 		{
@@ -794,7 +795,7 @@ YY_RULE_SETUP
 #line 49 "rd_parser/rd_scanner.ll"
 ECHO;
 	YY_BREAK
-#line 798 "rd_parser/rd_scanner.cc"
+#line 799 "rd_parser/rd_scanner.cc"
 case YY_STATE_EOF(INITIAL):
 	yyterminate();
 
@@ -981,7 +982,7 @@ static int yy_get_next_buffer (void)
 
 	else
 		{
-			int num_to_read =
+			yy_size_t num_to_read =
 			YY_CURRENT_BUFFER_LVALUE->yy_buf_size - number_to_move - 1;
 
 		while ( num_to_read <= 0 )
@@ -995,7 +996,7 @@ static int yy_get_next_buffer (void)
 
 			if ( b->yy_is_our_buffer )
 				{
-				int new_size = b->yy_buf_size * 2;
+				yy_size_t new_size = b->yy_buf_size * 2;
 
 				if ( new_size <= 0 )
 					b->yy_buf_size += b->yy_buf_size / 8;
@@ -1026,7 +1027,7 @@ static int yy_get_next_buffer (void)
 
 		/* Read in more data. */
 		YY_INPUT( (&YY_CURRENT_BUFFER_LVALUE->yy_ch_buf[number_to_move]),
-			(yy_n_chars), (size_t) num_to_read );
+			(yy_n_chars), num_to_read );
 
 		YY_CURRENT_BUFFER_LVALUE->yy_n_chars = (yy_n_chars);
 		}
@@ -1148,7 +1149,7 @@ static int yy_get_next_buffer (void)
 
 		else
 			{ /* need more input */
-			int offset = (yy_c_buf_p) - (yytext_ptr);
+			yy_size_t offset = (yy_c_buf_p) - (yytext_ptr);
 			++(yy_c_buf_p);
 
 			switch ( yy_get_next_buffer(  ) )
@@ -1172,7 +1173,7 @@ static int yy_get_next_buffer (void)
 				case EOB_ACT_END_OF_FILE:
 					{
 					if ( rd_yywrap( ) )
-						return EOF;
+						return 0;
 
 					if ( ! (yy_did_buffer_switch_on_eof) )
 						YY_NEW_FILE;
@@ -1424,7 +1425,7 @@ void rd_yypop_buffer_state (void)
  */
 static void rd_yyensure_buffer_stack (void)
 {
-	int num_to_alloc;
+	yy_size_t num_to_alloc;
     
 	if (!(yy_buffer_stack)) {
 
@@ -1521,12 +1522,11 @@ YY_BUFFER_STATE rd_yy_scan_string (yyconst char * yystr )
  * 
  * @return the newly allocated buffer state object.
  */
-YY_BUFFER_STATE rd_yy_scan_bytes  (yyconst char * yybytes, int  _yybytes_len )
+YY_BUFFER_STATE rd_yy_scan_bytes  (yyconst char * yybytes, yy_size_t  _yybytes_len )
 {
 	YY_BUFFER_STATE b;
 	char *buf;
-	yy_size_t n;
-	int i;
+	yy_size_t n, i;
     
 	/* Get memory for full buffer, including space for trailing EOB's. */
 	n = _yybytes_len + 2;
@@ -1608,7 +1608,7 @@ FILE *rd_yyget_out  (void)
 /** Get the length of the current token.
  * 
  */
-int rd_yyget_leng  (void)
+yy_size_t rd_yyget_leng  (void)
 {
         return rd_yyleng;
 }
diff --git a/serial.cmd b/serial.cmd
new file mode 100644
index 0000000..22b916b
--- /dev/null
+++ b/serial.cmd
@@ -0,0 +1,15 @@
+# serial job
+#PBS -l nodes=1:ppn=1,walltime=4:00:00
+# xxx PBS -l mem=80gb
+#
+# sends mail if the process aborts, when it begins, and
+# when it ends (abe)
+#PBS -m abe
+#
+# load intel compiler settings before running the program
+# since we compiled it with intel 10.1
+# module load intel/10.1
+# go to the directory with the program
+cd $HOME/src/knotkit
+# and run it
+LD_LIBRARY_PATH=/u/cseed/llvm-3.1/lib ./main
diff --git a/simplify_chain_complex.h b/simplify_chain_complex.h
new file mode 100644
index 0000000..189e48d
--- /dev/null
+++ b/simplify_chain_complex.h
@@ -0,0 +1,249 @@
+
+template<class R> class simplified_complex_generators
+{
+  unsigned new_n;
+  ptr<const module<R> > C;
+  basedvector<unsigned, 1> new_C_to_C_generator;
+  
+public:
+  simplified_complex_generators (const simplified_complex_generators &g)
+    : new_n(g.new_n),
+      C(g.C),
+      new_C_to_C_generator(g.new_C_to_C_generator)
+  { }
+  simplified_complex_generators (unsigned new_n_,
+				 ptr<const module<R> > C_,
+				 basedvector<unsigned, 1> new_C_to_C_generator_)
+    : new_n(new_n_),
+      C(C_),
+      new_C_to_C_generator(new_C_to_C_generator_)
+  { }
+  ~simplified_complex_generators () { }
+  
+  simplified_complex_generators &operator = (const simplified_complex_generators &); // doesn't exist
+  
+  unsigned dim () const { return new_n; }
+  unsigned free_rank () const { return new_n; }
+  grading generator_grading (unsigned i) const { return C->generator_grading (new_C_to_C_generator[i]); }
+  void show_generator (unsigned i) const { C->show_generator (new_C_to_C_generator[i]); }
+  R generator_ann (unsigned i) const { abort (); }
+};
+
+template<class R> 
+class chain_complex_simplifier
+{
+ public:
+  ptr<const module<R> > C;
+  unsigned n; // |C|
+  mod_map<R> d;
+  
+  ptr<const module<R> > new_C;
+  mod_map<R> new_d;
+  
+  // pi : C -> new_C
+  mod_map<R> pi;
+  
+  // iota : new_C -> C
+  mod_map<R> iota;
+  
+ private:
+  set<unsigned> canceled;
+  
+  basedvector<R, 1> cancel_binv;
+  basedvector<unsigned, 1> cancel_j;
+  basedvector<linear_combination<R>, 1> cancel_di;
+  
+  basedvector<linear_combination<R>, 1> new_d_columns;
+  basedvector<set<unsigned>, 1> preim;
+  
+  basedvector<linear_combination<R>, 1> iota_columns;
+  
+  void cancel (unsigned i, R b, unsigned j);
+  
+ public:
+  chain_complex_simplifier (ptr<const module<R> > C_,
+			    const mod_map<R> &d_,
+			    maybe<int> dh, maybe<int> dq);
+};
+
+template<class R> void
+chain_complex_simplifier<R>::cancel (unsigned i, R b, unsigned j)
+{
+  assert (i != j);
+  assert (b.is_unit ());
+  
+  R binv = b.recip ();
+  
+  canceled.push (i);
+  canceled.push (j);
+  
+  new_d_columns[i].yank (j);
+  preim[j].yank (i);
+  
+  for (linear_combination_const_iter<R> k = new_d_columns[i]; k; k ++)
+    preim[k.key ()].yank (i);
+  for (set_const_iter<unsigned> k = preim[i]; k; k ++)
+    new_d_columns[k.val ()].yank (i);
+  for (linear_combination_const_iter<R> k = new_d_columns[j]; k; k ++)
+    preim[k.key ()].yank (j);
+  
+  for (set_const_iter<unsigned> kk = preim[j]; kk; kk ++)
+    {
+      unsigned k = kk.val ();
+      R a = new_d_columns[k](j);
+      assert (a != 0);
+      
+      iota_columns[k].mulsub (a * binv, iota_columns[i]);
+    }
+  
+  iota_columns[i].clear ();
+  iota_columns[j].clear ();
+  
+  for (set_const_iter<unsigned> kk = preim[j]; kk; kk ++)
+    {
+      unsigned k = kk.val ();
+      R a = new_d_columns[k](j);
+      assert (a != 0);
+      
+      R abinv = a * binv;
+      
+      for (linear_combination_const_iter<R> ll = new_d_columns[i]; ll; ll ++)
+	{
+	  unsigned ell = ll.key ();
+	  R c = ll.val ();
+	  
+	  assert (! (canceled % k));
+	  assert (! (canceled % ell));
+	  assert (k != i);
+	  assert (k != j);
+	  assert (ell != i);
+	  assert (ell != j);
+	  assert (ell != k);
+	  
+	  new_d_columns[k].mulsub (abinv * c, ell);
+	  if (new_d_columns[k] % ell)
+	    preim[ell] += k;
+	  else
+	    preim[ell] -= k;
+	}
+    }
+  
+  for (set_const_iter<unsigned> k = preim[j]; k; k ++)
+    new_d_columns[k.val ()].yank (j);
+  
+  cancel_binv.append (binv);
+  cancel_j.append (j);
+  cancel_di.append (new_d_columns[i]);
+  
+  new_d_columns[i] = linear_combination<R> ();
+  preim[i].clear ();
+  new_d_columns[j].clear ();
+  preim[j].clear ();
+}
+
+template<class R>
+chain_complex_simplifier<R>::chain_complex_simplifier (ptr<const module<R> > C_,
+						       const mod_map<R> &d_,
+						       maybe<int> dh, maybe<int> dq)
+  : C(C_), n(C_->dim ()), d(d_),
+    new_d_columns(n),
+    preim(n),
+    iota_columns(n)
+{
+  for (unsigned i = 1; i <= n; i ++)
+    {
+      new_d_columns[i] = d.column_copy (i);
+      
+      for (linear_combination_const_iter<R> j = new_d_columns[i]; j; j ++)
+	preim[j.key ()].push (i);
+      
+      linear_combination<R> x (C);
+      x.muladd (1, i);
+      iota_columns[i] = x;
+    }
+  
+  for (unsigned i = n; i >= 1; i --)
+    {
+      if (canceled % i)
+	continue;
+      
+      grading igr = C->generator_grading (i);
+      for (linear_combination_const_iter<R> j = new_d_columns[i]; j; j ++)
+	{
+	  grading jgr = C->generator_grading (j.key ());
+	  if (j.val ().is_unit ()
+	      && (dh.is_none ()
+		  || (jgr.h - igr.h == dh.some ()))
+	      && (dq.is_none ()
+		  || (jgr.q - igr.q == dq.some ())))
+	    {
+	      cancel (i, j.val (), j.key ());
+	      break;
+	    }
+	}
+    }
+  
+  // ??? might not be completely simplified
+  
+  unsigned new_n = n - canceled.card ();
+  basedvector<unsigned, 1> new_C_to_C_generator (new_n),
+    C_to_new_C_generator (n);
+  for (unsigned i = 1, j = 1; i <= n; i ++)
+    {
+      if (canceled % i)
+	{
+	  C_to_new_C_generator[i] = 0;
+	  continue;
+	}
+      
+      C_to_new_C_generator[i] = j;
+      new_C_to_C_generator[j] = i;
+      j ++;
+    }
+  
+  new_C = (new base_module<R, simplified_complex_generators<R> >
+	   (simplified_complex_generators<R> (new_n, C, new_C_to_C_generator)));
+  map_builder<R> db (new_C);
+  
+  for (unsigned i = 1; i <= new_n; i ++)
+    {
+      unsigned i0 = new_C_to_C_generator[i];
+	  
+      for (linear_combination_const_iter<R> j0 = new_d_columns[i0]; j0; j0 ++)
+	{
+	  unsigned j = C_to_new_C_generator[j0.key ()];
+	  assert (j != 0);
+	  
+	  db[i].muladd (j0.val (), j);
+	}
+    }
+  new_d = mod_map<R> (db);
+  
+  map_builder<R> iotab (new_C, C);
+  
+  for (unsigned i = 1; i <= new_n; i ++)
+    iotab[i] = iota_columns[new_C_to_C_generator[i]];
+  iota = mod_map<R> (iotab);
+
+  map_builder<R> pib (C, new_C);
+  for (unsigned i = 1; i <= new_n; i ++)
+    pib[new_C_to_C_generator[i]].muladd (1, i);
+  
+  while (cancel_binv.size () > 0)
+    {
+      R binv = cancel_binv.pop ();
+      unsigned j = cancel_j.pop ();
+      linear_combination<R> di = cancel_di.pop ();
+      
+      for (linear_combination_const_iter<R> ll = di; ll; ll ++)
+	{
+	  R c = ll.val ();
+	  pib[j].mulsub (binv * c, pib[ll.key ()]);
+	}
+    }
+  pi = mod_map<R> (pib);
+  
+  assert (d.compose (iota) == iota.compose (new_d));
+  assert (new_d.compose (pi) == pi.compose (d));
+  assert (pi.compose (iota) == mod_map<R> (new_C, 1));
+}
diff --git a/spanning_tree_complex.h b/spanning_tree_complex.h
index 5a7a091..385a9a0 100644
--- a/spanning_tree_complex.h
+++ b/spanning_tree_complex.h
@@ -25,6 +25,9 @@ class spanning_tree_complex
 
   mod_map<R> totally_twisted_kh_d () const;
   mod_map<R> twisted_d2 () const;
+  mod_map<R> twisted_d2Un (unsigned n) const;
+  
+  mod_map<R> twisted_d2U1_test () const;
 };
 
 template<class F>
@@ -39,7 +42,7 @@ public:
   tree_generators (const spanning_tree_complex<F> &c_) : c(c_) { }
   ~tree_generators () { }
   
-  tree_generators &operator = (const tree_generators &); // doesn't exist
+  tree_generators &operator = (const tree_generators &) = delete;
   
   unsigned dim () const { return c.trees.size (); }
   unsigned free_rank () const { return c.trees.size (); }
@@ -210,6 +213,291 @@ spanning_tree_complex<F>::twisted_d2 () const
   return mod_map<R> (b);
 }
 
+template<class F> mod_map<fraction_field<polynomial<F> > >
+spanning_tree_complex<F>::twisted_d2Un (unsigned n) const
+{
+  assert (kd.marked_edge);
+  assert (n >= 1);
+  
+  map_builder<R> b (C);
+  
+  basedvector<int, 1> edge_weight (kd.num_edges ());
+  for (unsigned i = 1; i <= kd.num_edges (); i ++)
+    edge_weight[i] = 1;  // (1 << i);
+  
+  int total_weight = 0;
+  for (unsigned i = 1; i <= kd.num_edges (); i ++)
+    total_weight += edge_weight[i];
+  
+  int shift = 2 * total_weight * n;
+  
+  for (unsigned i = 1; i <= trees.size (); i ++)
+    {
+      set<unsigned> t = trees[i];
+      
+      smallbitset r (kd.n_crossings);
+      for (unsigned k = 1; k <= kd.n_crossings; k ++)
+	{
+	  if ((edge_height[k] == 1) == (t % k))
+	    r.push (k);
+	}
+      smoothing s (kd, r);
+      
+      for (set_const_iter<unsigned> ee = t; ee; ee ++)
+	{
+	  unsigned e = ee.val ();
+	  
+	  if (edge_height[e] != 0)
+	    continue;
+	  
+	  for (unsigned f = 1; f <= bg.num_edges (); f ++)
+	    {
+	      if (edge_height[f] != 1 || (t % f))
+		continue;
+	      
+	      set<unsigned> t2 (COPY, t);
+	      t2.yank (e);
+	      t2.push (f);
+	      unsigned j = tree_idx(t2, 0);
+	      if (j == 0)
+		continue;
+	      
+	      bool neg = 0;
+	      for (unsigned a = kd.marked_edge;;)
+		{
+		  unsigned p = s.edge_to (kd, a);
+		  if (kd.ept_crossing[p] == e)
+		    {
+		      if (s.is_crossing_from_ept (kd, r, p))
+			neg = s.crossing_from_inside (kd, r, e);
+		      else
+			neg = s.crossing_to_inside (kd, r, e);
+		      break;
+		    }
+		  else if (kd.ept_crossing[p] == f)
+		    {
+		      if (s.is_crossing_from_ept (kd, r, p))
+			neg = s.crossing_from_inside (kd, r, f);
+		      else
+			neg = s.crossing_to_inside (kd, r, f);
+		      break;
+		    }
+		  
+                  a = s.next_edge (kd, r, a);
+		  assert (a != kd.marked_edge);
+		}
+	      
+	      set<unsigned> neither (COPY, t);
+	      neither.yank (e);
+	      
+	      smallbitset neither_r (kd.n_crossings);
+	      for (unsigned k = 1; k <= kd.n_crossings; k ++)
+		{
+		  if ((edge_height[k] == 1) == (neither % k))
+		    neither_r.push (k);
+		}
+	      smoothing neither_s (kd, neither_r);
+	      
+	      set<unsigned> both (COPY, t);
+	      both.push (f);
+	      
+	      int A = 0;
+	      for (unsigned k = 1; k <= kd.num_edges (); k ++)
+		{
+		  if (neither_s.edge_circle[k]
+		      != neither_s.edge_circle[kd.marked_edge])
+		    A += edge_weight[k];
+		}
+	      
+	      smallbitset both_r (kd.n_crossings);
+	      for (unsigned k = 1; k <= kd.n_crossings; k ++)
+		{
+		  if ((edge_height[k] == 1) == (both % k))
+		    both_r.push (k);
+		}
+	      smoothing both_s (kd, both_r);
+	      
+	      int B = 0;
+	      for (unsigned k = 1; k <= kd.num_edges (); k ++)
+		{
+		  if (both_s.edge_circle[k]
+		      != both_s.edge_circle[kd.marked_edge])
+		    B += edge_weight[k];
+		}
+	      
+	      assert (A >= 0 && B >= 0);
+	      
+	      R x;
+	      
+	      for (int k = 1; k <= (int)n; k ++)
+		{
+		  if (n % k == 0)
+		    {
+		      int l = n / k;
+		      assert (k * l == (int)n);
+		      
+		      assert (shift >= k*A + l*B);
+		      
+		      if (neg) // ^ (bool)((n + 1) & 1))
+			{
+			  x += R (polynomial<F> (1, (unsigned)(shift + k*A + l*B)));
+			  x += R (polynomial<F> (1, (unsigned)(shift - k*A - l*B)));
+			}
+		      else
+			{
+			  x += R (polynomial<F> (1, (unsigned)(shift - k*A + l*B)));
+			  x += R (polynomial<F> (1, (unsigned)(shift + k*A - l*B)));
+			}
+		    }
+		}
+	      
+	      b[i].muladd (x, j);
+	    }
+	}
+    }
+  
+  return mod_map<R> (b);
+}
+
+template<class F> mod_map<fraction_field<polynomial<F> > >
+spanning_tree_complex<F>::twisted_d2U1_test () const
+{
+  assert (kd.marked_edge);
+  
+  map_builder<R> b (C);
+  
+  basedvector<int, 1> edge_weight (kd.num_edges ());
+  for (unsigned i = 1; i <= kd.num_edges (); i ++)
+    edge_weight[i] = 1;  // (1 << i);
+  
+  int total_weight = 0;
+  for (unsigned i = 1; i <= kd.num_edges (); i ++)
+    total_weight += edge_weight[i];
+  
+  int shift = 2 * total_weight;
+  
+  for (unsigned i = 1; i <= trees.size (); i ++)
+    {
+      set<unsigned> t = trees[i];
+      
+      smallbitset r (kd.n_crossings);
+      for (unsigned k = 1; k <= kd.n_crossings; k ++)
+	{
+	  if ((edge_height[k] == 1) == (t % k))
+	    r.push (k);
+	}
+      smoothing s (kd, r);
+      
+      for (set_const_iter<unsigned> ee = t; ee; ee ++)
+	{
+	  unsigned e = ee.val ();
+	  
+	  if (edge_height[e] != 0)
+	    continue;
+	  
+	  for (unsigned f = 1; f <= bg.num_edges (); f ++)
+	    {
+	      if (edge_height[f] != 1 || (t % f))
+		continue;
+	      
+	      set<unsigned> t2 (COPY, t);
+	      t2.yank (e);
+	      t2.push (f);
+	      unsigned j = tree_idx(t2, 0);
+	      if (j == 0)
+		continue;
+	      
+	      bool neg = 0;
+	      for (unsigned a = kd.marked_edge;;)
+		{
+		  unsigned p = s.edge_to (kd, a);
+		  if (kd.ept_crossing[p] == e)
+		    {
+		      if (s.is_crossing_from_ept (kd, r, p))
+			neg = s.crossing_from_inside (kd, r, e);
+		      else
+			neg = s.crossing_to_inside (kd, r, e);
+		      break;
+		    }
+		  else if (kd.ept_crossing[p] == f)
+		    {
+		      if (s.is_crossing_from_ept (kd, r, p))
+			neg = s.crossing_from_inside (kd, r, f);
+		      else
+			neg = s.crossing_to_inside (kd, r, f);
+		      break;
+		    }
+		  
+                  a = s.next_edge (kd, r, a);
+		  assert (a != kd.marked_edge);
+		}
+	      
+	      set<unsigned> neither (COPY, t);
+	      neither.yank (e);
+	      
+	      smallbitset neither_r (kd.n_crossings);
+	      for (unsigned k = 1; k <= kd.n_crossings; k ++)
+		{
+		  if ((edge_height[k] == 1) == (neither % k))
+		    neither_r.push (k);
+		}
+	      smoothing neither_s (kd, neither_r);
+	      
+	      set<unsigned> both (COPY, t);
+	      both.push (f);
+	      
+	      int A = 0;
+	      for (unsigned k = 1; k <= kd.num_edges (); k ++)
+		{
+		  if (neither_s.edge_circle[k]
+		      != neither_s.edge_circle[kd.marked_edge])
+		    A += edge_weight[k];
+		}
+	      
+	      smallbitset both_r (kd.n_crossings);
+	      for (unsigned k = 1; k <= kd.n_crossings; k ++)
+		{
+		  if ((edge_height[k] == 1) == (both % k))
+		    both_r.push (k);
+		}
+	      smoothing both_s (kd, both_r);
+	      
+	      int B = 0;
+	      for (unsigned k = 1; k <= kd.num_edges (); k ++)
+		{
+		  if (both_s.edge_circle[k]
+		      != both_s.edge_circle[kd.marked_edge])
+		    B += edge_weight[k];
+		}
+	      
+	      assert (A >= 0 && B >= 0);
+	      
+	      printf ("A = %d, B = %d\n", A, B);
+	      
+	      R x;
+
+	      // worked:
+	      // bad: -, -/+, +
+	      assert (shift >= A && shift >= B);
+	      if (neg)
+		{
+		  x += R (polynomial<F> (1, (unsigned)(shift + A)));
+		  x += R (polynomial<F> (1, (unsigned)(shift - B)));
+		}
+	      else
+		{
+		  x += R (polynomial<F> (1, (unsigned)(shift - A)));
+		  x += R (polynomial<F> (1, (unsigned)(shift + B)));
+		}
+	      
+	      b[i].muladd (x, j);
+	    }
+	}
+    }
+  
+  return mod_map<R> (b);
+}
+
 template<class F> mod_map<fraction_field<polynomial<F> > >
 spanning_tree_complex<F>::totally_twisted_kh_d () const
 {
diff --git a/sseq.cpp b/sseq.cpp
index 662c8ab..0a1c4aa 100644
--- a/sseq.cpp
+++ b/sseq.cpp
@@ -2,7 +2,8 @@
 #include <knotkit.h>
 
 sseq_page::sseq_page (const sseq_bounds &b)
-  : rank(b.width ()),
+  : k(0),
+    rank(b.width ()),
     im_rank(b.width ())
 {
   for (unsigned i = 0; i < b.width (); i ++)
@@ -140,87 +141,6 @@ sseq_page::delta_poincare_polynomial (const sseq_bounds &b) const
   return r;
 }
 
-void
-sseq_page::texshow (FILE *fp, const sseq_bounds &b, unsigned dh, bool last)
-{
-  fprintf (fp, "\
-\\begin{tikzpicture}[scale=.66]\n\
-  \\draw[->] (0,0) -- (%d.5,0) node[right] {$t$};\n\
-  \\draw[->] (0,0) -- (0,%d.5) node[above] {$q$};\n\
-  \\draw[step=1] (0,0) grid (%d,%d);\n",
-	   b.width (), (b.height () + 1) / 2, b.width (), (b.height () + 1) / 2);
-
-  fprintf (fp, "  \\draw (%.3f,-0.8) node[below] {$",
-	   ((double)b.width () + 0.5) / 2);
-  if (last)
-    fprintf (fp, "E_\\infty = ");
-  fprintf (fp, "E_{%d}$};\n",
-	   dh);
-  
-  /* label axes */
-  for (int x = b.minh; x <= b.maxh; x ++)
-    {
-      fprintf (fp, "\
-  \\draw (%d.5,-.2) node[below] {$%d$};\n",
-	       x - b.minh, x);
-    }
-  for (int y = b.minq; y <= b.maxq; y += 2)
-    {
-      fprintf (fp, "\
-  \\draw (-.2,%d.5) node[left] {$%d$};\n",
-	       (y - b.minq) / 2, y);
-    }
-  
-  for (unsigned i = 0; i < b.width (); i ++)
-    for (unsigned j = 0; j < b.height (); j ++)
-      {
-	int r = rank[i][j];
-	if (r == 1)
-	  {
-	    assert (is_even (j));
-	    fprintf (fp, "\
-  \\fill (%d.5, %d.5) circle (.15);\n",
-		     i, j / 2);
-	  }
-	else if (r > 1)
-	  {
-	    assert (is_even (j));
-	    fprintf (fp, "\
-  \\draw (%d.5, %d.5) node {$%d$};\n",
-		    i, j / 2, r);
-	  }
-      }
-  
-  for (unsigned i = 0; i < b.width (); i ++)
-    for (unsigned j = 0; j < b.height (); j ++)
-      {
-	unsigned r = im_rank[i][j];
-	if (r == 0)
-	  continue;
-	
-	unsigned dx = dh,
-	  dy = dh - 1;  // in "boxes"
-	assert (dx >= 2); // eliminated d_1
-	
-	double h = sqrt ((double)(dx * dx + dy * dy));
-	double xadj = (double)dx / h * 0.3;
-	double yadj = (double)dy / h * 0.3;
-	
-	assert (is_even (j));
-	fprintf (fp, "  \\draw[->] (%.3f, %.3f) -- ",
-		(double)i + 0.5 + xadj, (double)(j / 2) + 0.5 + yadj);
-	if (r > 1)
-	  fprintf (fp, "node[color=red!75!black] {$%d$} ", r);
-	fprintf (fp, "(%.3f, %.3f);\n",
-		(double)(i + dx) + 0.5 - xadj, (double)(j / 2 + dy) + 0.5 - yadj);
-      }
-  
-  fprintf (fp, "\
-\\end{tikzpicture}\n");
-  
-  fflush (fp);
-}
-
 sseq
 sseq::operator + (const sseq &ss2) const
 {
@@ -280,36 +200,6 @@ sseq::shift (int dh, int dq) const
 	       pages);
 }
 
-bool
-sseq::equal_as_spaces (const sseq &ss) const
-{
-  if (bounds != ss.bounds
-      || pages.size () != ss.pages.size ())
-    return 0;
-  
-  for (unsigned i = 1; i <= pages.size (); i ++)
-    {
-      if (!pages[i].equal_as_spaces (ss.pages[i]))
-	return 0;
-    }
-  
-  return 1;
-}
-
-void
-sseq::texshow (FILE *fp, std::string name)
-{
-  fprintf (fp, "\\section{Knot $%s$}\n", name.c_str ());
-  
-  unsigned n = pages.size ();
-  for (unsigned i = 1; i <= n; i ++)
-    fprintf (fp, "$\\rank E_%d = %d$ \\\\\n",
-	    i + 1, pages[i].total_rank ());
-  
-  for (unsigned i = 1; i <= n; i ++)
-    pages[i].texshow (fp, bounds, i + 1, i == n);
-}
-
 void
 sseq_builder::cancel (unsigned d)
 {
diff --git a/sseq.h b/sseq.h
index 29fa5a9..85cfd6a 100644
--- a/sseq.h
+++ b/sseq.h
@@ -7,6 +7,27 @@ class sseq_bounds
   
  public:
   sseq_bounds () { }
+  
+  template<class R, class M> sseq_bounds (ptr<const module<R> > C, M m)
+  {
+    grading hq1 = m(C->generator_grading (1));
+    minh = maxh = hq1.h;
+    minq = maxq = hq1.q;
+    
+    for (unsigned i = 2; i <= C->dim (); i ++)
+      {
+	grading hq = m(C->generator_grading (i));
+	if (hq.h < minh)
+	  minh = hq.h;
+	if (hq.h > maxh)
+	  maxh = hq.h;
+	if (hq.q < minq)
+	  minq = hq.q;
+	if (hq.q > maxq)
+	  maxq = hq.q;
+      }
+  }
+  
   sseq_bounds (int minh_, int maxh_, int minq_, int maxq_)
     : minh(minh_), maxh(maxh_), minq(minq_), maxq(maxq_)
   { }
@@ -34,25 +55,40 @@ class sseq_bounds
     && minq == b.minq
     && maxq == b.maxq;
   }
+  bool operator != (const sseq_bounds &b) const { return !operator == (b); }
 };
 
 class sseq_page
 {
  public:
   /* bounds come from sseq */
+  unsigned k;
+  grading dk_gr;
   vector<vector<unsigned> > rank,
     im_rank;
   
  public:
   sseq_page () { }
   sseq_page (const sseq_bounds &b);
+  template<class R, class M> sseq_page (const sseq_bounds &b,
+					unsigned k_,
+					grading dk_gr_,
+					mod_map<R> d,
+					M m);
   sseq_page (const sseq_page &pg)
-    : rank(pg.rank), im_rank(pg.im_rank)
+    : k(pg.k), dk_gr(pg.dk_gr), rank(pg.rank), im_rank(pg.im_rank)
   { }
   ~sseq_page () { }
   
-  bool operator == (const sseq_page &pg) const { return rank == pg.rank && im_rank == pg.im_rank; }
-  bool equal_as_spaces (const sseq_page &pg) const { return rank == pg.rank; }
+  bool operator == (const sseq_page &pg) const
+  {
+    return k == pg.k
+    && dk_gr == pg.dk_gr
+    && rank == pg.rank
+    && im_rank == pg.im_rank;
+  }
+
+  bool operator != (const sseq_page &pg) const { return !operator == (pg); }
   
   unsigned total_rank () const;
   multivariate_laurentpoly<Z> poincare_polynomial (const sseq_bounds &b) const;
@@ -64,7 +100,7 @@ class sseq_page
 		     const sseq_bounds &b1, const sseq_page &pg1,
 		     const sseq_bounds &b2, const sseq_page &pg2);
   
-  void texshow (FILE *fp, const sseq_bounds &b, unsigned dh, bool last);
+  template<class M> void texshow (FILE *fp, const sseq_bounds &b, const char *label, M m);
 };
 
 class sseq
@@ -75,8 +111,10 @@ class sseq
   
  public:
   sseq () { }
-  sseq (const sseq_bounds &b, const basedvector<sseq_page, 1> &pages_)
-    : bounds(b), pages(pages_)
+  sseq (const sseq_bounds &b,
+	const basedvector<sseq_page, 1> &pages_)
+    : bounds(b),
+      pages(pages_)
   { }
   sseq (const sseq_bounds &b)
     : bounds(b)
@@ -84,7 +122,12 @@ class sseq
   sseq (const sseq &ss) : bounds(ss.bounds), pages(ss.pages) { }
   ~sseq () { }
   
-  sseq &operator = (const sseq &ss) { bounds = ss.bounds; pages = ss.pages; return *this; }
+  sseq &operator = (const sseq &ss)
+  {
+    bounds = ss.bounds;
+    pages = ss.pages;
+    return *this;
+  }
   
   sseq operator + (const sseq &ss2) const;  // direct sum
   sseq otimes (const sseq &ss2) const; // tensor product
@@ -95,279 +138,157 @@ class sseq
   unsigned homological_width () const { return pages[1].homological_width (bounds); }
   
   bool operator == (const sseq &ss) const { return bounds == ss.bounds && pages == ss.pages; }
-  bool equal_as_spaces (const sseq &ss) const;
+  bool operator != (const sseq &ss) const { return !operator == (ss); }
   
-  void texshow (FILE *fp, std::string name);
+  template<class M> void texshow (FILE *fp, M m);
 };
 
-template<class R> class simplified_complex_generators
+template<class R, class M> 
+sseq_page::sseq_page (const sseq_bounds &b,
+		      unsigned k_,
+		      grading dk_gr_,
+		      mod_map<R> dk,
+		      M m)
+  : k(k_),
+    dk_gr(dk_gr_),
+    rank(b.width ()),
+    im_rank(b.width ())
 {
-  unsigned new_n;
-  ptr<const module<R> > C;
-  basedvector<unsigned, 1> new_C_to_C_generator;
+  dk.check_grading (dk_gr);
   
-public:
-  simplified_complex_generators (const simplified_complex_generators &g)
-    : new_n(g.new_n),
-      C(g.C),
-      new_C_to_C_generator(g.new_C_to_C_generator)
-  { }
-  simplified_complex_generators (unsigned new_n_,
-				 ptr<const module<R> > C_,
-				 basedvector<unsigned, 1> new_C_to_C_generator_)
-    : new_n(new_n_),
-      C(C_),
-      new_C_to_C_generator(new_C_to_C_generator_)
-  { }
-  ~simplified_complex_generators () { }
+  grading m_dk_gr = m.map_delta(dk_gr);
   
-  simplified_complex_generators &operator = (const simplified_complex_generators &); // doesn't exist
-  
-  unsigned dim () const { return new_n; }
-  unsigned free_rank () const { return new_n; }
-  grading generator_grading (unsigned i) const { return C->generator_grading (new_C_to_C_generator[i]); }
-  void show_generator (unsigned i) const { C->show_generator (new_C_to_C_generator[i]); }
-  R generator_ann (unsigned i) const { abort (); }
-};
-
-template<class R> 
-class chain_complex_simplifier
-{
- public:
-  typedef typename R::linear_combination linear_combination;
-  typedef typename R::linear_combination_const_iter linear_combination_const_iter;
-  
- public:
-  ptr<const module<R> > C;
-  unsigned n; // |C|
-  const mod_map<R> &d;
-  
-  ptr<const module<R> > new_C;
-  mod_map<R> new_d;
-  
-  // chain homotopy equivalences
-  // pi : C -> new_C, iota : new_C -> C
-  mod_map<R> pi, iota;
-  
- private:
-  basedvector<linear_combination, 1> new_d0;
-  basedvector<set<unsigned>, 1> preim;
-  
-  bool build_pi_iota;
-  mod_map<R> pi0, iota0;
-  set<unsigned> canceled;
-  
-  void cancel (unsigned i, R b, unsigned j);
-  
- public:
-  chain_complex_simplifier (ptr<const module<R> > C_,
-			    const mod_map<R> &d_,
-			    int dh,
-			    bool build_pi_iota_ = 0);
-  
-};
-
-template<class R> void
-chain_complex_simplifier<R>::cancel (unsigned i, R b, unsigned j)
-{
-  assert (i != j);
-  assert (b.is_unit ());
-  
-  R binv = b.recip ();
-  
-  canceled.push (i);
-  canceled.push (j);
-  
-  new_d0[i].yank (j);
-  preim[j].yank (i);
-  
-  for (linear_combination_const_iter k = new_d0[i]; k; k ++)
-    preim[k.key ()].yank (i);
-  for (set_const_iter<unsigned> k = preim[i]; k; k ++)
-    new_d0[k.val ()].yank (i);
-  for (linear_combination_const_iter k = new_d0[j]; k; k ++)
-    preim[k.key ()].yank (j);
-
-  if (build_pi_iota)
+  for (unsigned i = 0; i < b.width (); i ++)
     {
-      mod_map<R> local_pi0 (C, 1),
-	local_iota0 (C, 1);
+      vector<unsigned> r (b.height ()),
+	im_r (b.height ());
+      for (unsigned j = 0; j < b.height (); j ++)
+	r[j] = im_r[j] = 0;
       
-      local_pi0[i].clear ();
-      local_pi0[j].clear ();
-      
-      local_iota0[i].clear ();
-      local_iota0[j].clear ();
-      
-      for (linear_combination_const_iter ll = new_d0[i]; ll; ll ++)
-	{
-	  R c = ll.val ();
-	  local_pi0[j].mulsub (binv * c, ll.key ());
-	}
-      
-      for (set_const_iter<unsigned> kk = preim[j]; kk; kk ++)
-	{
-	  unsigned k = kk.val ();
-	  R a = new_d0[k](j);
-	  assert (a != 0);
-	  
-	  local_iota0[k].mulsub (a * binv, i);
-	}
-      
-      pi0 = local_pi0.compose (pi0);
-      iota0 = iota0.compose (local_iota0);
+      rank[i] = r;
+      im_rank[i] = im_r;
     }
   
-  for (set_const_iter<unsigned> kk = preim[j]; kk; kk ++)
+  ptr<const module<R> > C = dk.domain ();
+  for (unsigned i = 1; i <= C->dim (); i ++)
     {
-      unsigned k = kk.val ();
-      R a = new_d0[k](j);
-      assert (a != 0);
-      
-      R abinv = a * binv;
-      
-      for (linear_combination_const_iter ll = new_d0[i]; ll; ll ++)
-	{
-	  unsigned ell = ll.key ();
-	  R c = ll.val ();
-	  
-	  assert (! (canceled % k));
-	  assert (! (canceled % ell));
-	  assert (k != i);
-	  assert (k != j);
-	  assert (ell != i);
-	  assert (ell != j);
-	  assert (ell != k);
-	  
-	  new_d0[k].mulsub (abinv * c, ell);
-	  if (new_d0[k] % ell)
-	    preim[ell] += k;
-	  else
-	    preim[ell] -= k;
-	}
+      grading gr = m(C->generator_grading (i));
+      rank[(unsigned)(gr.h - b.minh)][(unsigned)(gr.q - b.minq)] ++;
     }
   
-  for (set_const_iter<unsigned> k = preim[j]; k; k ++)
-    new_d0[k.val ()].yank (j);
-  
-  new_d0[i].clear ();
-  preim[i].clear ();
-  new_d0[j].clear ();
-  preim[j].clear ();
-  
-#if 0  // expensive
-  if (build_pi_iota)
+  ptr<const free_submodule<R> > im = dk.image ();
+  for (unsigned i = 1; i <= im->dim (); i ++)
     {
-      assert (pi0.compose (d) == new_d0.compose (pi0));
-      assert (iota0.compose (new_d0) == d.compose (iota0));
+      grading gr = m(im->generator_grading (i));
+      im_rank[(unsigned)(gr.h - m_dk_gr.h - b.minh)][(unsigned)(gr.q - m_dk_gr.q - b.minq)] ++;
     }
-#endif
 }
 
-template<class R>
-chain_complex_simplifier<R>::chain_complex_simplifier (ptr<const module<R> > C_,
-						       const mod_map<R> &d_,
-						       int dh,
-						       bool build_pi_iota_)
-  : C(C_), n(C_->dim ()), d(d_),
-    new_d0(COPY2, d_.explicit_columns ()),
-    preim(C_->dim ()),
-    build_pi_iota(build_pi_iota_)
+
+template<class M> void
+sseq_page::texshow (FILE *fp, const sseq_bounds &b, const char *label, M m)
 {
+  grading m_dk_gr = m.map_delta(dk_gr);
+  
+  unsigned dh = m_dk_gr.h,
+    dq = m_dk_gr.q;
+  
+  fprintf (fp, "\
+\\begin{tikzpicture}[scale=.66]\n\
+  \\draw[->] (0,0) -- (%d.5,0) node[right] {$t$};\n\
+  \\draw[->] (0,0) -- (0,%d.5) node[above] {$q$};\n\
+  \\draw[step=1] (0,0) grid (%d,%d);\n",
+	   b.width (), b.height (), b.width (), b.height ());
+  
+  fprintf (fp, "  \\draw (%.3f,-0.8) node[below] {%s};\n",
+	   ((double)b.width () + 0.5) / 2, label);
+  
+  /* label axes */
+  for (int x = b.minh; x <= b.maxh; x ++)
+    {
+      fprintf (fp, "\
+  \\draw (%d.5,-.2) node[below] {$",
+	       x - b.minh);
+      m.x_label (fp, x);
+      fprintf (fp, "$};\n");
+    }
+  for (int y = b.minq; y <= b.maxq; y ++)
+    {
+      fprintf (fp, "\
+  \\draw (-.2,%d.5) node[left] {$",
+	       y - b.minq);
+      m.y_label (fp, y);
+      fprintf (fp, "$};\n");
+    }
+  
+  for (unsigned i = 0; i < b.width (); i ++)
+    for (unsigned j = 0; j < b.height (); j ++)
+      {
+	int r = rank[i][j];
+	if (r == 1)
+	  {
+	    fprintf (fp, "\
+  \\fill (%d.5, %d.5) circle (.15);\n",
+		     i, j);
+	  }
+	else if (r > 1)
+	  {
+	    fprintf (fp, "\
+  \\draw (%d.5, %d.5) node {$%d$};\n",
+		    i, j, r);
+	  }
+      }
+  
+  for (unsigned i = 0; i < b.width (); i ++)
+    for (unsigned j = 0; j < b.height (); j ++)
+      {
+	unsigned r = im_rank[i][j];
+	if (r == 0)
+	  continue;
+	
+	int dx = dh,
+	  dy = dq;  // in "boxes"
+	
+	double h = sqrt ((double)(dx * dx + dy * dy));
+	double xadj = (double)dx / h * 0.3;
+	double yadj = (double)dy / h * 0.3;
+	
+	fprintf (fp, "  \\draw[->] (%.3f, %.3f) -- ",
+		(double)i + 0.5 + xadj, (double)j + 0.5 + yadj);
+	if (r > 1)
+	  fprintf (fp, "node[color=red!75!black] {$%d$} ", r);
+	fprintf (fp, "(%.3f, %.3f);\n",
+		 (double)((int)i + dx) + 0.5 - xadj, (double)((int)j + dy) + 0.5 - yadj);
+      }
+  
+  fprintf (fp, "\
+\\end{tikzpicture}\n");
+  
+  fflush (fp);
+}
+
+template<class M> void
+sseq::texshow (FILE *fp, M m)
+{
+  unsigned n = pages.size ();
   for (unsigned i = 1; i <= n; i ++)
     {
-      for (linear_combination_const_iter j = d[i]; j; j ++)
-	preim[j.key ()].push (i);
+      if (i == n)
+	fprintf (fp, "$\\rank E_\\infty = \\rank E_%d = %d$ \\\\\n",
+		 pages[i].k, pages[i].total_rank ());
+      else
+	fprintf (fp, "$\\rank E_%d = %d$ \\\\\n",
+		 pages[i].k, pages[i].total_rank ());
     }
   
-  if (build_pi_iota)
+  for (unsigned i = 1; i <= n; i ++)
     {
-      pi0 = mod_map<R> (C, 1);
-      iota0 = mod_map<R> (C, 1);
-      
-#if 0 // expensive
-      assert (pi0.compose (d) == new_d0.compose (pi0));
-      assert (iota0.compose (new_d0) == d.compose (iota0));
-#endif
-    }
-  
- Lretry:
-  for (unsigned i = n; i >= 1; i --)
-    {
-      if (canceled % i)
-	continue;
-      
-      grading igr = C->generator_grading (i);
-      for (linear_combination_const_iter j = new_d0[i]; j; j ++)
-	{
-	  grading jgr = C->generator_grading (j.key ());
-	  assert (jgr.h >= igr.h);
-	  if (j.val ().is_unit ()
-	      && jgr.h - igr.h == dh)
-	    {
-	      cancel (i, j.val (), j.key ());
-	      goto Lretry;
-	    }
-	}
-    }
-  
-  unsigned new_n = n - canceled.card ();
-  basedvector<unsigned, 1> new_C_to_C_generator (new_n),
-    C_to_new_C_generator (n);
-  for (unsigned i = 1, j = 1; i <= n; i ++)
-    {
-      if (canceled % i)
-	{
-	  C_to_new_C_generator[i] = 0;
-	  continue;
-	}
-      
-      C_to_new_C_generator[i] = j;
-      new_C_to_C_generator[j] = i;
-      j ++;
-    }
-  
-  new_C = (new base_module<R, simplified_complex_generators<R> >
-	   (simplified_complex_generators<R> (new_n, C, new_C_to_C_generator)));
-  new_d = mod_map<R> (new_C);
-  
-  for (unsigned i = 1; i <= new_n; i ++)
-    {
-      unsigned i0 = new_C_to_C_generator[i];
-	  
-      for (linear_combination_const_iter j0 = new_d0[i0]; j0; j0 ++)
-	{
-	  unsigned j = C_to_new_C_generator[j0.key ()];
-	  assert (j != 0);
-	  
-	  new_d[i].muladd (j0.val (), j);
-	}
-    }
-  
-  if (build_pi_iota)
-    {
-      pi = mod_map<R> (C, new_C);
-      iota = mod_map<R> (new_C, C);
-      
-      for (unsigned i0 = 1; i0 <= n; i0 ++)
-	{
-	  for (linear_combination_const_iter j0 = pi0[i0]; j0; j0 ++)
-	    {
-	      unsigned j = C_to_new_C_generator[j0.key ()];
-	      pi[i0].muladd (j0.val (), j);
-	    }
-	}
-      
-      for (unsigned i = 1; i <= new_n; i ++)
-	{
-	  unsigned i0 = new_C_to_C_generator[i];
-	  iota[i] = iota0[i0];
-	}
-      
-#if 0  // expensive
-      assert (pi.compose (d) == new_d.compose (pi));
-      assert (d.compose (iota) == iota.compose (new_d));
-#endif
+      char buf[1000];
+      if (i == n)
+	sprintf (buf, "$E_\\infty = E_{%d}$", pages[i].k);
+      else
+	sprintf (buf, "$E_{%d}$", pages[i].k);
+      pages[i].texshow (fp, bounds, buf, m);
     }
 }
 
diff --git a/steenrod_square.cpp b/steenrod_square.cpp
new file mode 100644
index 0000000..7376a69
--- /dev/null
+++ b/steenrod_square.cpp
@@ -0,0 +1,547 @@
+
+#include <knotkit.h>
+
+steenrod_square::steenrod_square (const cube<Z2> &cor_,
+				  mod_map<Z2> &d_,
+				  const chain_complex_simplifier<Z2> &s_)
+  : cor(cor_), d(d_), s(s_)
+{
+  for (unsigned i = 1; i <= cor.khC->dim (); i ++)
+    {
+      grading igr = cor.khC->generator_grading (i);
+      KGij[igr].push (i);
+    }
+}
+
+Z2
+steenrod_square::sC (unsigned x, unsigned y) const
+{
+  pair<unsigned, unsigned> x_sm = cor.generator_state_monomial (x),
+    y_sm = cor.generator_state_monomial (y);
+  
+  unsigned changed = x_sm.first ^ y_sm.first;
+  assert (unsigned_bitcount (changed) == 1);
+  unsigned i = unsigned_ffs (changed);
+  
+  unsigned k = 0;
+  for (unsigned j = 1; j < i; j ++)
+    {
+      if (unsigned_bittest (x_sm.first, j))
+	k ++;
+    }
+  
+  return Z2 (k);
+}
+
+Z2
+steenrod_square::fC (unsigned x, unsigned y) const
+{
+  pair<unsigned, unsigned> x_sm = cor.generator_state_monomial (x),
+    y_sm = cor.generator_state_monomial (y);
+  
+  unsigned changed = x_sm.first ^ y_sm.first;
+  assert (unsigned_bitcount (changed) == 2);
+  
+  unsigned i = unsigned_ffs (changed),
+    j = unsigned_ffs (unsigned_bitclear (changed, i));
+  assert (i < j);
+  
+  unsigned ki = 0;
+  for (unsigned l = 1; l < i; l ++)
+    {
+      if (unsigned_bittest (x_sm.first, l))
+	ki ++;
+    }
+  unsigned kj = 0;
+  for (unsigned l = i + 1; l < j; l ++)
+    {
+      if (unsigned_bittest (x_sm.first, l))
+	kj ++;
+    }
+  
+  return Z2 (ki) * Z2 (kj);
+}
+
+triple<set<unsigned>, // G1cx
+       map<unsigned, unsigned>, // bx
+       map<unsigned, Z2> > // sx
+steenrod_square::boundary_matching (grading cgr,
+				    linear_combination<Z2> c,
+				    unsigned x) const
+{
+  set<unsigned> G1cx;
+  map<unsigned, unsigned> bx;
+  map<unsigned, Z2> sx;
+  
+  for (linear_combination_const_iter<Z2> yy = c; yy; yy ++)
+    {
+      assert (yy.val () == 1);
+      unsigned y = yy.key ();
+      
+      if (d[y](x) == 1)
+	G1cx.push (y);
+    }
+  assert (is_even (G1cx.card ()));
+  
+  for (set_const_iter<unsigned> i = G1cx; i; i ++)
+    {
+      unsigned y = i.val ();
+      i ++;
+      unsigned bxy = i.val ();
+      
+      bx.push (y, bxy);
+      bx.push (bxy, y);
+      
+      sx.push (y, 0);
+      sx.push (bxy, Z2 (1) + sC (x, y) + sC (x, bxy));
+    }
+  
+#ifndef NDEBUG
+  for (set_const_iter<unsigned> yy = G1cx; yy; yy ++)
+    {
+      unsigned y = yy.val ();
+      assert (sx(y) + sx(bx(y)) == Z2 (1) + sC (x, y) + sC (x, bx(y)));
+    }
+#endif
+  
+  return triple<set<unsigned>, // G1cx
+		map<unsigned, unsigned>, // bx
+		map<unsigned, Z2> > // sx
+    (G1cx, bx, sx);
+}
+
+mod_map<Z2>
+steenrod_square::sq1 () const
+{
+  map_builder<Z2> b (s.new_C);
+  
+  for (unsigned i = 1; i <= s.new_C->dim (); i ++)
+    {
+      grading cgr = s.new_C->generator_grading (i);
+      linear_combination<Z2> c = s.iota[i];
+      assert (d.map (c) == 0);
+      
+      grading gr2 (cgr.h + 1, cgr.q);
+      if (! (KGij % gr2))
+	continue;
+      
+      linear_combination<Z2> sq1c (cor.khC);
+      
+      for (set_const_iter<unsigned> x = KGij[gr2]; x; x ++)
+	{
+	  triple<set<unsigned>,
+		 map<unsigned, unsigned>,
+		 map<unsigned, Z2> > t = boundary_matching (cgr, c, x.val ());
+	  
+	  Z2 a = 0;
+	  for (set_const_iter<unsigned> y = t.first; y; y ++)
+	    a += t.third(y.val ());
+	      
+	  sq1c.muladd (a, x.val ());
+	}
+      // display ("sq1c:\n", sq1c);
+      assert (d.map (sq1c) == 0);
+      
+      b[i].muladd (1, s.pi.map (sq1c));
+    }
+  
+  return mod_map<Z2> (b);
+}
+
+set<pair<unsigned, unsigned> > 
+steenrod_square::make_G2cx (grading cgr,
+			    linear_combination<Z2> c,
+			    unsigned x) const
+{
+  set<pair<unsigned, unsigned> > G2cx;
+  
+  for (linear_combination_const_iter<Z2> yy = c; yy; yy ++)
+    {
+      unsigned y = yy.key ();
+      
+      for (linear_combination_const_iter<Z2> zz = d[y]; zz; zz ++)
+	{
+	  unsigned z = zz.key ();
+	  
+	  if (d[z](x) == 1)
+	    G2cx.push (pair<unsigned, unsigned> (z, y));
+	}
+    }
+  return G2cx;
+}
+
+class graph
+{
+public:
+  unsigned n_vertices;
+  unsigned n_edges;
+  
+  basedvector<unsigned, 1> z;
+  basedvector<unsigned, 1> y;
+  map<pair<unsigned, unsigned>, unsigned> zy_vertex;
+  
+  unsigned vertex (unsigned z, unsigned y) const
+  {
+    return zy_vertex(pair<unsigned, unsigned> (z, y));
+  }
+  
+  basedvector<unsigned, 1> edge_from, edge_to;
+  set<unsigned> edge_oriented;
+  basedvector<Z2, 1> edge_label;
+  
+  map<unsigned, set<unsigned> > incident_edges;
+  
+public:
+  graph (set<pair<unsigned, unsigned> > G2cx);
+  ~graph () { }
+  
+  void add_edge (unsigned from_z, unsigned from_y,
+		 unsigned to_z, unsigned to_y,
+		 Z2 label,
+		 bool oriented);
+  
+  unsigned num_components () const;
+  Z2 f () const; // sum of weights
+  Z2 g () const;
+};
+
+graph::graph (set<pair<unsigned, unsigned> > G2cx)
+  : n_vertices(G2cx.card ()),
+    n_edges(0),
+    z(n_vertices),
+    y(n_vertices)
+{
+  unsigned j = 1;
+  for (set_const_iter<pair<unsigned, unsigned> > i = G2cx; i; i ++, j ++)
+    {
+      z[j] = i.val ().first;
+      y[j] = i.val ().second;
+      zy_vertex.push (i.val (), j);
+    }
+#ifndef NDEBUG
+  for (unsigned i = 1; i <= n_vertices; i ++)
+    {
+      assert (i == vertex (z[i], y[i]));
+    }
+#endif
+  assert (j == n_vertices + 1);
+}
+
+void
+graph::add_edge (unsigned from_z, unsigned from_y,
+		 unsigned to_z, unsigned to_y,
+		 Z2 label,
+		 bool oriented)
+{
+  unsigned e = ++ n_edges;
+  
+  unsigned from = vertex (from_z, from_y),
+    to = vertex (to_z, to_y);
+  
+  edge_from.append (from);
+  edge_to.append (to);
+  edge_label.append (label);
+  assert (e == edge_label.size ());
+  
+  if (oriented)
+    edge_oriented.push (e);
+  
+  incident_edges[from].push (e);
+  incident_edges[to].push (e);
+}
+
+unsigned
+graph::num_components () const
+{
+  unionfind<1> u (n_vertices);
+  for (unsigned i = 1; i <= n_edges; i ++)
+    u.join (edge_from[i], edge_to[i]);
+  return u.num_sets ();
+}
+
+Z2
+graph::f () const
+{
+  Z2 a = 0;
+  for (unsigned i = 1; i <= n_edges; i ++)
+    a += edge_label[i];
+  return a;
+}
+
+Z2
+graph::g () const
+{
+  set<unsigned> done;
+  
+  set<unsigned> A,
+    B;
+  for (unsigned i = 1; i <= n_edges; i ++)
+    {
+      if (done % i)
+	continue;
+      
+      unsigned j = i;
+      unsigned v = edge_to[j];
+      for (;;)
+	{
+	  if (edge_oriented % j)
+	    {
+	      if (edge_to[j] == v)
+		A.push (j);
+	      else
+		B.push (j);
+	    }
+	  done.push (j);
+	  
+	  if (incident_edges[v].head () == j)
+	    j = incident_edges[v].tail ();
+	  else
+	    {
+	      assert (j == incident_edges[v].tail ());
+	      j = incident_edges[v].head ();
+	    }
+	  if (v == edge_to[j])
+	    v = edge_from[j];
+	  else
+	    {
+	      assert (v == edge_from[j]);
+	      v = edge_to[j];
+	    }
+	  
+	  if (j == i)
+	    {
+	      assert (v == edge_to[i]);
+	      break;
+	    }
+	}
+    }
+  
+  assert (Z2 (A.card ()) == Z2 (B.card ()));
+  return Z2 (A.card ());
+}
+
+pair<set<unsigned>,
+     map<unsigned, unsigned> > 
+steenrod_square::ladybug_matching (unsigned x, unsigned y) const
+{
+  set<unsigned> Gxy;
+  map<unsigned, unsigned> lxy;
+  
+  for (linear_combination_const_iter<Z2> zz = d[y]; zz; zz ++)
+    {
+      unsigned z = zz.key ();
+      
+      if (d[z](x) == 1)
+	Gxy.push (z);
+    }
+  
+  if (Gxy.card () == 2)
+    {
+      unsigned z1 = Gxy.head (),
+	z2 = Gxy.tail ();
+      assert (z1 != z2);
+      
+      lxy.push (z1, z2);
+      lxy.push (z2, z1);
+    }
+  else if (Gxy.card () == 4)
+    {
+      pair<unsigned, unsigned> x_sm = cor.generator_state_monomial (x),
+	y_sm = cor.generator_state_monomial (y);
+      
+      unsigned changed = x_sm.first ^ y_sm.first;
+      assert (unsigned_bitcount (changed) == 2);
+      unsigned i = unsigned_ffs (changed),
+	j = unsigned_ffs (unsigned_bitclear (changed, i));
+      
+      unsigned s00 = y_sm.first;
+      unsigned s01 = unsigned_bitset (s00, i);
+      unsigned s10 = unsigned_bitset (s00, j);
+      
+      smoothing from_s (cor.kd, smallbitset (cor.n_crossings, s00));
+      
+      basedvector<unsigned, 1> from_circle_edge_rep (from_s.n_circles);
+      for (unsigned j = 1; j <= cor.kd.num_edges (); j ++)
+	from_circle_edge_rep[from_s.edge_circle[j]] = j;
+      
+      unsigned e1 = cor.kd.ept_edge (cor.kd.crossings[i][1]),
+	e3 = cor.kd.ept_edge (cor.kd.crossings[i][3]);
+      
+      unsigned p = from_s.edge_circle[e1];
+      assert (p == from_s.edge_circle[e3]);
+      assert (unsigned_bittest (y_sm.second, p));  // p+
+      
+      smoothing conf01 (cor.kd, smallbitset (cor.n_crossings, s01)),
+	conf10 (cor.kd, smallbitset (cor.n_crossings, s10));
+      
+      assert (conf01.edge_circle[e1] != conf01.edge_circle[e3]);
+      assert (conf10.edge_circle[e1] != conf10.edge_circle[e3]);
+      
+      unsigned m01 = 0,
+	m10 = 0;
+      for (unsigned i = 1; i <= from_s.n_circles; i ++)
+	{
+	  if (unsigned_bittest (y_sm.second, i))
+	    {
+	      m01 = unsigned_bitset (m01, conf01.edge_circle[from_circle_edge_rep[i]]);
+	      m10 = unsigned_bitset (m10, conf10.edge_circle[from_circle_edge_rep[i]]);
+	    }
+	}
+      
+      unsigned m01_a = unsigned_bitclear (unsigned_bitset (m01,
+							   conf01.edge_circle[e1]),
+					  conf01.edge_circle[e3]),
+	m10_a = unsigned_bitclear (unsigned_bitset (m10,
+						    conf10.edge_circle[e1]),
+				   conf10.edge_circle[e3]);
+      unsigned z01_a = cor.generator (s01, m01_a),
+	z10_a = cor.generator (s10, m10_a);
+      assert (Gxy(z01_a) && Gxy(z10_a));
+      lxy.push (z01_a, z10_a);
+      lxy.push (z10_a, z01_a);
+      
+      unsigned m01_b = unsigned_bitclear (unsigned_bitset (m01,
+							   conf01.edge_circle[e3]),
+					  conf01.edge_circle[e1]),
+	m10_b = unsigned_bitclear (unsigned_bitset (m10,
+						    conf10.edge_circle[e3]),
+				   conf10.edge_circle[e1]);
+      unsigned z01_b = cor.generator (s01, m01_b),
+	z10_b = cor.generator (s10, m10_b);
+      assert (Gxy(z01_b) && Gxy(z10_b));
+      lxy.push (z01_b, z10_b);
+      lxy.push (z10_b, z01_b);
+    }
+  else
+    assert (Gxy.card () == 0);
+  
+  assert (Gxy.card () == lxy.card ());
+  
+  return pair<set<unsigned>, map<unsigned, unsigned> > (Gxy, lxy);
+}
+
+Z2
+steenrod_square::sq2_coeff (grading cgr,
+			    linear_combination<Z2> c,
+			    unsigned x) const
+{
+  set<pair<unsigned, unsigned> > G2cx = make_G2cx (cgr, c, x);
+  
+  graph G (G2cx);
+  
+  set<unsigned> ys;
+  for (unsigned i = 1; i <= G.n_vertices; i ++)
+    ys += G.y[i];
+  
+  for (set_const_iter<unsigned> yy = ys; yy; yy ++)
+    {
+      unsigned y = yy.val ();
+      
+      pair<set<unsigned>,
+	   map<unsigned, unsigned> > p = ladybug_matching (x, y);
+      
+      set<unsigned> Gxy = p.first;
+      map<unsigned, unsigned> lxy = p.second;
+      
+      for (set_const_iter<unsigned> zz = Gxy; zz; zz ++)
+	{
+	  unsigned z = zz.val ();
+	  unsigned lxyz = lxy(z);
+	  if (z < lxyz)
+	    {
+	      G.add_edge (z, y,
+			  lxyz, y,
+			  fC (x, y), 0);
+	    }
+	}
+    }
+  
+  set<unsigned> zs;
+  for (unsigned i = 1; i <= G.n_vertices; i ++)
+    zs += G.z[i];
+  
+  for (set_const_iter<unsigned> zz = zs; zz; zz ++)
+    {
+      unsigned z = zz.val ();
+      triple<set<unsigned>,
+	     map<unsigned, unsigned>,
+	     map<unsigned, Z2> > t = boundary_matching (cgr, c, z);
+      
+      set<unsigned> Gcz = t.first;
+      map<unsigned, unsigned> bz = t.second;
+      map<unsigned, Z2> sz = t.third;
+      
+      for (set_const_iter<unsigned> yy = Gcz; yy; yy ++)
+	{
+	  unsigned y = yy.val ();
+	  
+	  unsigned bzy = bz(y);
+	  if (y < bzy)
+	    {
+	      if (sz(y) == 0
+		  && sz(bzy) == 1)
+		{
+		  G.add_edge (z, y,
+			      z, bzy,
+			      0, 1);
+		}
+	      else if (sz(y) == 1
+		       && sz(bzy) == 0)
+		{
+		  G.add_edge (z, bzy,
+			      z, y,
+			      0, 1);
+		}
+	      else
+		{
+		  assert (sz(y) == sz(bzy));
+		  G.add_edge (z, y,
+			      z, bzy,
+			      0, 0);
+		}
+	    }
+	}
+    }
+  
+#ifndef NDEBUG
+  for (unsigned i = 1; i <= G.n_edges; i ++)
+    {
+      assert (G.incident_edges[i].card () == 2);
+    }
+#endif
+  
+  // printf ("G.n_edges = %d\n", G.n_edges);
+  // printf ("#|G| = %d\n", G.num_components ());
+  
+  return (Z2 (G.num_components ())
+	  + G.f ()
+	  + G.g ());
+}
+
+mod_map<Z2>
+steenrod_square::sq2 () const
+{
+  map_builder<Z2> b (s.new_C);
+  
+  for (unsigned i = 1; i <= s.new_C->dim (); i ++)
+    {
+      grading cgr = s.new_C->generator_grading (i);
+      linear_combination<Z2> c = s.iota[i];
+      assert (d.map (c) == 0);
+      
+      grading gr2 (cgr.h + 2, cgr.q);
+      if (! (KGij % gr2))
+	continue;
+      
+      linear_combination<Z2> sq2c (cor.khC);
+      for (set_const_iter<unsigned> x = KGij[gr2]; x; x ++)
+	sq2c.muladd (sq2_coeff (cgr, c, x.val ()), x.val ());
+      
+      // display ("sq2c:\n", sq2c);
+      assert (d.map (sq2c) == 0);
+      
+      b[i].muladd (1, s.pi.map (sq2c));
+    }
+  
+  return mod_map<Z2> (b);
+}
diff --git a/steenrod_square.h b/steenrod_square.h
new file mode 100644
index 0000000..176c5ce
--- /dev/null
+++ b/steenrod_square.h
@@ -0,0 +1,42 @@
+
+class steenrod_square
+{
+  // cube of resolutions (in steenrod_square, c will denote a cycle)
+  const cube<Z2> &cor;
+  
+  mod_map<Z2> d;
+  const chain_complex_simplifier<Z2> &s;
+  
+  map<grading, set<unsigned> > KGij;
+  
+  // sC, fC take generators as input, not hypercube vertices
+  Z2 sC (unsigned x, unsigned y) const;
+  Z2 fC (unsigned x, unsigned y) const;
+  
+  pair<set<unsigned>, // Gxy
+      map<unsigned, unsigned>  // lxy
+    > ladybug_matching (unsigned x, unsigned y) const;
+  
+  triple<set<unsigned>, // Gcx
+      map<unsigned, unsigned>, // bx
+      map<unsigned, Z2> // sx
+    > boundary_matching (grading cgr,
+			 linear_combination<Z2> c,
+			 unsigned x) const;
+  
+  set<pair<unsigned, unsigned> > make_G2cx (grading cgr,
+					    linear_combination<Z2> c,
+					    unsigned x) const;
+  Z2 sq2_coeff (grading cgr,
+		linear_combination<Z2> c,
+		unsigned x) const;
+  
+ public:
+  steenrod_square (const cube<Z2> &cor_,
+		   mod_map<Z2> &d_,
+		   const chain_complex_simplifier<Z2> &s_);
+  ~steenrod_square () { }
+  
+  mod_map<Z2> sq1 () const;
+  mod_map<Z2> sq2 () const;
+};
diff --git a/todo.txt b/todo.txt
index ad38695..bf0f287 100644
--- a/todo.txt
+++ b/todo.txt
@@ -13,10 +13,8 @@ in knotkit/
    deepcopy
 
 in lib/
- - add hashset
  - between interoperation between sets that require bounds (bitset,
    ullmanset) and those that don't (set, hashset)
- - add make_pair, etc.
 
 in algebra/
  - linear_combnation can be more efficient (eg no searching when
@@ -25,3 +23,73 @@ in algebra/
  - cleaner interface for gcd, etc. in fraction_field
  - support kernel, cokernel of torsion modules
  - change gcd et al so gcd is always positive
+ - split linear_combination into internal and external versions
+
+git:
+ - fix bug in Josh's version
+ - make this master, merge into Josh's version
+ - merge in square to master => then write paper!
+ - merge in other improvements (sped up linear algebra?  multivariate (laurent) polynomial?)
+ - should exponents on polynomials be Z or unsigned?
+
+general:
+ - unify simplify_chain_complex, sseq, kernel/image/homology (as much as possible)
+ - sseq over Z
+ - quadratic linear solver (enumerator?)
+ - classes should either COPY or fully SHARE, but not some of each
+   (e.g. knot_diagram, resolution_diagram, etc.)  -- this is the right
+   semantics.
+ - clean up/delete setcommon, mapcommon
+ - incorporate sage dump into standard classes
+ - update use of tuples in steenrod square
+
+ - test mutation invariance, other questions of Robert and Sucharit
+ - write paper
+
+c++11:
+ - make vector wrap std::vector (probably faster)
+ 
+ - use tuple instead of pair, triple; and use tie, a great idiom
+ - standardize ring, field interface so test_ring works for everything
+ - get clear on default semantics: prefer default when possible
+ 
+ - standardize use of {} for initializers
+ - initializer_list for container types
+ - support for iterator
+
+ - compress Steenrod square data
+ - get mpimain running again, run 15 xing mutants
+
+ - simplifications to run (5, 8) with symmetries.
+
+notes:
+
+all rings (R): multivariate_laurentpoly<R>
+
+fields (F): Z2 Zp Q fraction_field<E> (for just pids?)
+
+euclidean domains (E): Z polynomial<F>
+
+euclidean domain interface:
+
+num/demon : denom must be a unit
+ = num.operator / (denom)
+
+denom.divides (num)
+denom | num
+
+  num.divide_exact (denom)
+
+(q, r) = num.divide_with_remainder (denom)
+  (q, r) = num / denom
+  (q, r) = num.operator / (denom)
+satisfies r = 0 or f(r) < f(denom) and num = q*denom + r
+
+ gcd (a, b)
+ lcm (a, b)
+ r = num.mod (denom)
+satisfies r = 0 or f(r) < f(denom)
+
+
+(d, s, t) = a.extended_gcd (b)
+ satifies d is the gcd and d = a*s + b*t