Added compute_forgetful_signs to main. Set up mpimain to compute

forgetful spectral sequence over Z/97 for all links with 14 or fewer
crossings.

cube_impl.h

@@ -365,6 +365,13 @@ cube<R>::compute_dinv (unsigned c)
       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);
@@ -398,12 +405,12 @@ cube<R>::compute_dinv (unsigned c)
 		  j2 = unsigned_bitset (j2, x);
 		  j2 = unsigned_bitclear (j2, y);
 		  
-		  p[generator (i, j)].muladd (1, generator (i2, j2));
+		  p[generator (i, j)].muladd (sign, generator (i2, j2));
 		  
 		  j2 = unsigned_bitclear (j2, x);
 		  j2 = unsigned_bitset (j2, y);
 		  
-		  p[generator (i, j)].muladd (1, generator (i2, j2));
+		  p[generator (i, j)].muladd (sign, generator (i2, j2));
 		}
 	      else
 		{
@@ -411,7 +418,7 @@ cube<R>::compute_dinv (unsigned c)
 		  j2 = unsigned_bitclear (j2, x);
 		  j2 = unsigned_bitclear (j2, y);
 		  
-		  p[generator (i, j)].muladd (1, generator (i2, j2));
+		  p[generator (i, j)].muladd (sign, generator (i2, j2));
 		}
 	    }
 	  else
@@ -425,7 +432,7 @@ cube<R>::compute_dinv (unsigned c)
 		  // 1 -> 1
 		  j2 = unsigned_bitset (j2, x);
 		  
-		  p[generator (i, j)].muladd (1, generator (i2, j2));
+		  p[generator (i, j)].muladd (sign, generator (i2, j2));
 		}
 	      else if ((unsigned_bittest (j, a)
 			&& !unsigned_bittest (j, b))
@@ -435,7 +442,7 @@ cube<R>::compute_dinv (unsigned c)
 		  // a, b -> x
 		  j2 = unsigned_bitclear (j2, x);
 		  
-		  p[generator (i, j)].muladd (1, generator (i2, j2));
+		  p[generator (i, j)].muladd (sign, generator (i2, j2));
 		}
 	    }
 	}

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 ++)
@@ -521,6 +521,63 @@ 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 += std::abs (lk);
+      }
+  
+  return total_lk;
+}
+
 knot_diagram::knot_diagram (connect_sum,
 			    const knot_diagram &d1,
 			    const knot_diagram &d2)

knot_diagram.h

@@ -157,6 +157,8 @@ 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;

knot_tables.cpp

@@ -1,7 +1,7 @@
 
 #include <knotkit.h>
 
-#define HOME "/u/cseed/src/knotkit/"
+#define HOME "/Users/cotton/src/knotkit/"
 
 bool verbose = 0;
 

main.cpp

@@ -849,6 +849,183 @@ test_forgetful_ss ()
       }
 }
 
+void
+test_forgetful_signs ()
+{
+  typedef Zp<97> R;
+  
+#if 0
+  for (unsigned i = 1; i <= 14; i ++)
+    for (unsigned j = 1; j <= mt_links (i, 0); j ++)
+      {
+	knot_diagram kd (mt_link (i, 0, j));
+      }
+#endif
+#if 1
+  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");
+      }
+}
+
 /* Kh homotopy type:
    - check for CP^2,
    - mutation invariance
@@ -1578,6 +1755,9 @@ convert_15 ()
 int
 main ()
 {
+  test_forgetful_signs ();
+  return 0;
+  
   find_width4_in_h0 ();
   // convert_15 ();
   // test_knot_sq ();

mpimain.cpp

@@ -14,11 +14,24 @@ master ()
 {
   basedvector<knot_desc, 1> work;
   
+  for (unsigned i = 1; i <= 14; i ++)
+    for (unsigned j = 1; j <= mt_links (i, 0); j ++)
+      {
+	knot_diagram kd (mt_links (i, 0, 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 --)
@@ -87,6 +100,198 @@ master ()
     }
 }
 
+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;
+}
+
+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);
+}
+
 void
 slave ()
 {
@@ -106,40 +311,13 @@ slave ()
 	    
 	    printf ("[% 2d] CMD_DO %s\n", rank, desc.name ().c_str ());
 	    
+	    assert (desc.t == knot_desc::MT);
 	    char buf[1000];
-	    sprintf (buf, "/scratch/network/cseed/incoming/K%d_%d.dat.gz",
+	    sprintf (buf, "/scratch/network/cseed/forgetful/L%d_%d.dat.gz",
 		     desc.i, desc.j);
 	    
-	    struct stat stat_buf;
+	    if (! file_exits ())
-	    if (stat (buf, &stat_buf) != 0)
+	      compute_forgetful (rank, desc, buf);
-	      {
-		if (errno == ENOENT)
-		  {
-		    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);
-		    
-		    gzfile_writer w (buf);
-		    write (w, sq1);
-		    write (w, sq2);
-		  }
-		else
-		  {
-		    stderror ("stat: %s", buf);
-		    exit (EXIT_FAILURE);
-		  }
-	      }
 	    else
 	      printf ("skip %s: exists.\n", buf);
 	    

simplify_chain_complex.h

@@ -63,7 +63,7 @@ class chain_complex_simplifier
  public:
   chain_complex_simplifier (ptr<const module<R> > C_,
 			    const mod_map<R> &d_,
-			    int dh);
+			    int dq);
   
 };
 
@@ -145,7 +145,7 @@ chain_complex_simplifier<R>::cancel (unsigned i, R b, unsigned j)
 template<class R>
 chain_complex_simplifier<R>::chain_complex_simplifier (ptr<const module<R> > C_,
 						       const mod_map<R> &d_,
-						       int dh)
+						       int dq)
   : C(C_), n(C_->dim ()), d(d_),
     new_d_columns(n),
     preim(n),
@@ -172,9 +172,8 @@ chain_complex_simplifier<R>::chain_complex_simplifier (ptr<const module<R> > C_,
       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
+	      && jgr.q - igr.q == dq
 	      )
 	    {
 	      cancel (i, j.val (), j.key ());