diff --git a/Makefile b/Makefile
index 402a664..6be5ece 100644
--- a/Makefile
+++ b/Makefile
@@ -19,7 +19,8 @@ CXXFLAGS = $(OPTFLAGS) -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
@@ -38,7 +39,8 @@ 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 -lz
 
diff --git a/algebra/Z2.h b/algebra/Z2.h
index 0cb8799..3765fb9 100644
--- a/algebra/Z2.h
+++ b/algebra/Z2.h
@@ -11,6 +11,7 @@ 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) { }
diff --git a/algebra/linear_combination.h b/algebra/linear_combination.h
index c3d6bba..a53727a 100644
--- a/algebra/linear_combination.h
+++ b/algebra/linear_combination.h
@@ -40,7 +40,7 @@ class linear_combination
       v.push (i.key (), R (COPY, i.val ()));
   }
   ~linear_combination () { }
-
+  
   linear_combination &operator = (const linear_combination &lc)
   {
     m = lc.m;
diff --git a/algebra/module.h b/algebra/module.h
index 71f3eff..63cdb87 100644
--- a/algebra/module.h
+++ b/algebra/module.h
@@ -628,7 +628,8 @@ class map_impl : public refcounted
   
   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 +658,11 @@ 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 +672,7 @@ 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 +682,7 @@ 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 +705,7 @@ 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 +726,7 @@ 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 +761,7 @@ 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);
     
@@ -884,8 +889,10 @@ 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
@@ -1637,8 +1644,8 @@ 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> > 
diff --git a/cube.h b/cube.h
index 49bd26e..2583cab 100644
--- a/cube.h
+++ b/cube.h
@@ -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;
diff --git a/knotkit.h b/knotkit.h
index c0165ae..bfa33db 100644
--- a/knotkit.h
+++ b/knotkit.h
@@ -14,11 +14,13 @@ 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>
 
 unsigned rolfsen_crossing_knots (unsigned n);
diff --git a/lib/pair.h b/lib/pair.h
index ce37dbb..46d8f51 100644
--- a/lib/pair.h
+++ b/lib/pair.h
@@ -54,8 +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;
@@ -81,13 +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);
@@ -95,3 +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/main.cpp b/main.cpp
index 9ba848a..1a7fc05 100644
--- a/main.cpp
+++ b/main.cpp
@@ -127,189 +127,143 @@ rank_lte (multivariate_laurentpoly<Z> p,
 int
 main ()
 {
-#if 0
-  ullmanset<1> empty;
-  assert (empty.size () == 0);
-  assert (empty.card () == 0);
-  assert (empty.is_empty ());
+  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);
   
-  ullmanset<1> empty2;
-  assert (empty == empty2);
+	  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);
 
-  ullmanset<1> emptycopy (COPY, empty);
-  assert (emptycopy == empty);
+	  mod_map<Z2> sq2 = sq.sq2 ();
+	  if (a)
+	    assert (sq2 == 0);
+	  else
+	    display ("sq2:\n", sq2);
   
-  ullmanset<1> s (50, {1,2,5,5,2,1,3,4});
-  assert (s.card () == 5);
-  
-  ullmanset<1> scopy (COPY, s);
-  assert (scopy == s);
-  
-  printf ("s:");
-  for (int x : s)
-    printf (" %d", x);
-  newline ();
-#endif
-  
-  map<unsigned, const char *> m = { { 5, "foo" }, { 11, "bar" } };
-  
-  hashmap<unsigned, const char *> m2 = { { 5, "foo" }, { 11, "bar" }, { 37, "baz" } };
-  for (std::pair<unsigned, const char *> i : m2)
-    {
-      printf ("%d -> %s\n", i.first, i.second);
-    }
+	  assert (sq1.compose (sq1) == 0);
+	  assert (sq2.compose (sq2) + sq1.compose (sq2).compose (sq1) == 0);
+	}
   
 #if 0
-  set<unsigned> s {1,2,5,5,2,1,3,4};
-  printf ("s:");
-  for (int x : s)
-    printf (" %d", x);
-  newline ();
-#endif
+  typedef Z2 R;
   
-#if 0
-  basedvector<int, 1> v (4);
-  v[1] = 2;
-  v[2] = -1;
-  v[3] = 0xbf32813;
-  v[4] = -352182;
-
-  vector<int> sleb (512);
-  for (unsigned i = 0; i < 512; i ++)
-    sleb[i] = ((int)i << 23);
-
-  printf ("sleb:\n");
-  for (unsigned i = 0; i < 512; i ++)
-    printf ("  %d\n", sleb[i]);
-  
-  vector<unsigned> uleb (512);
-  for (unsigned i = 0; i < 512; i ++)
-    uleb[i] = ((unsigned)i << 23);
-  
-  printf ("uleb:\n");
-  for (unsigned i = 0; i < 512; i ++)
-    printf ("  %u\n", uleb[i]);
-  
-  polynomial<Z> one (1);
-  polynomial<Z> x (1, 1);
-  polynomial<Z> bigp (Z (367521));
-  
-  polynomial<Z> t = x + bigp*one;
-  polynomial<Z> p = (t*t*t + t*t + t + one)*(bigp*t*t + bigp*bigp*t + bigp*bigp*bigp)*(bigp*bigp*t*t*t + bigp*t*t + t + one);
-  display ("p: ", p);
-  
-  {
-    file_writer w ("test.dat");
-    write (w, v);
-    write (w, p);
-    write (w, sleb);
-    write (w, uleb);
-  }
-  
-  {
-    gzfile_writer w ("test.2.dat.gz");
-    write (w, v);
-    write (w, p);
-    write (w, sleb);
-    write (w, uleb);
-  }
-  
-  system ("gzip -cd test.2.dat.gz > test.2.dat");
-  
-  {
-    file_reader r ("test.dat");
-    basedvector<int, 1> v2 (r);
-    polynomial<Z> p2 (r);
-    
-    vector<int> sleb2 (r);
-    printf ("sleb2:\n");
-    for (unsigned i = 0; i < 512; i ++)
-      printf ("  %d\n", sleb2[i]);
-    
-    vector<unsigned> uleb2 (r);
-    assert (v == v2);
-    assert (p == p2);
-    assert (sleb == sleb2);
-    assert (uleb == uleb2);
-  }
-  
-  system ("gzip -c9 test.dat > test.dat.gz");
-  
-  {
-    gzfile_reader r ("test.dat.gz");
-    basedvector<int, 1> v2 (r);
-    polynomial<Z> p2 (r);
-    vector<int> sleb2 (r);
-    vector<unsigned> uleb2 (r);
-    assert (v == v2);
-    assert (p == p2);
-    assert (sleb == sleb2);
-    assert (uleb == uleb2);
-  }
-#endif
-  
-#if 0
-  
-#if 0
-  test_ring<Z> (0);
-  test_ring<polynomial<Z> > (0);
-  
-  test_field<Z2> ();
-  test_field<Zp<7> > ();
-  test_field<Q> ();
-  // test_field<fraction_field<Zp<7> > > ();
-#endif
-  
-#if 0
-  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
-  for (unsigned i = 1; i <= 12; i ++)
+  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 1
+  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
+  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 ++)
+    {
+      printf ("P = "); display (i.key ());
+      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 ();
+	}
+    }
+#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/simplify_chain_complex.h b/simplify_chain_complex.h
new file mode 100644
index 0000000..ec330e3
--- /dev/null
+++ b/simplify_chain_complex.h
@@ -0,0 +1,248 @@
+
+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_,
+			    int dh);
+  
+};
+
+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_,
+						       int dh)
+  : 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 ());
+	  assert (jgr.h >= igr.h);
+	  if (j.val ().is_unit ()
+	      && jgr.h - igr.h == dh)
+	    {
+	      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 82abe9d..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>
@@ -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/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;
+};