knotkit/main.cpp

#include <knotkit.h>

// test for ring

template<class R> void
test_ring (int p)
{
  R zero (0);
  R one (1);
  R minus_one (-1);
  
  assert (zero == 0);
  assert (zero | zero);
  assert (one | zero);
  assert (minus_one | zero);
  assert (! (zero | one));
  assert (! (zero | minus_one));
  
  assert (one.is_unit ());
  assert (minus_one.is_unit ());
  assert (one.recip () == one);
  assert (minus_one.recip () == minus_one);
  
  if (p)
    assert (R (p) == 0);
  
  if (p != 2)
    assert (one != minus_one);
  
  int n = (p
	   ? std::min (p, 20)
	   : 20);
  for (int i = -n; i <= n; i ++)
    {
      R x (i);
      if (x.is_unit ())
	{
	  assert (x * x.recip () == one);
	  assert (x.recip () * x == one);
	  assert (x.recip ().recip () == x);
	}
      
      assert (one | x);
      assert (minus_one | x);
      
      if (x != 0)
	{
	  assert (x | zero);
	  assert (! (zero | x));
	}
      
      for (int j = -n; j <= n; j ++)
	{
	  R y (j);
	  
	  assert (- (-x) == x);
	  assert (x + y == y + x);
	  assert (x * y == y * x);
	  
	  if (x != 0 && x | y)
	    {
	      R q = y.divide_exact (x);
	      assert (y == q * x);
	    }
	  
	  if (x != 0 || y != 0)
	    {
	      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 ++)
	    {
	      R z (k);
	      
	      assert ((x + y) + z == x + (y + z));
	      assert ((x * y) * z == x * (y * z));
	      assert (x*(y + z) == x*y + x*z);
	      assert ((x + y)*z == x*z + y*z);
	    }
	}
    }
}

template<class F> void
test_field ()
{
  for (unsigned i = 1; i <= 8; i ++)
    for (unsigned j = 1; j <= rolfsen_crossing_knots (i); j ++)
      {
	knot_diagram kd (rolfsen_knot (i, j));
	
	show (kd); newline ();
	
	cube<F> c (kd);
	mod_map<F> d = c.compute_d (1, 0, 0, 0, 0);
	
	assert (d.compose (d) == 0);
	
	ptr<const quotient_module<F> > H = d.homology ();
	display ("H:\n", *H);
	
	chain_complex_simplifier<F> s (c.khC, d, 1);
	display ("s.new_C:\n", *s.new_C);
	
	assert (H->dim () == s.new_C->dim ());
      }
}

bool
rank_lte (multivariate_laurentpoly<Z> p,
	  multivariate_laurentpoly<Z> q)
{
  for (map<multivariate_laurent_monomial, Z>::const_iter i = p.coeffs; i; i ++)
    {
      Z a = i.val ();
      Z b = q.coeffs(i.key (), Z (0));
      assert (a != 0 && b != 0);
      
      if (a > b)
	return 0;
    }
  return 1;
}

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;
  
  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);
  
  reader r (buf);
  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 (hwidth == 2)
	continue;
      if (i.key ().t == knot_desc::MT
	  && i.key ().diagram ().num_components () == 1)
	continue;
      
      knot_kh_sq.push (i.key (), i.val ());
    }
  
  printf ("done.\n");
}

static const int block_size = 100;

void
show_st (map<knot_desc,
	     pair<mod_map<Z2>, mod_map<Z2> > > knot_kh_sq,
	 knot_desc desc)
{
  pair<mod_map<Z2>, mod_map<Z2> > p = knot_kh_sq(desc);
  
  mod_map<Z2> sq1 = p.first;
  mod_map<Z2> sq2 = p.second;
  
  printf ("%s  ", desc.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 ();
  
  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 ();
      
      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 (s1 != 0)
	{
	  if (first)
	    first = 0;
	  else
	    printf (", ");
	  printf ("(%d, %d) -> (%d, %d, %d, %d)",
		  hq.h, hq.q,
		  s1, s2, s3, s4);
	}
    }
  
  newline ();
}

void
sage_show (std::string prefix,
	   map<grading, basedvector<unsigned, 1> > hq_gens,
	   ptr<const module<Z2> > H)
{
  printf ("%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
	printf (", ");
      printf ("(%d, %d): %d", i.key ().h, i.key ().q, i.val ().size ());
    }      
  printf ("}\n");
}

void
sage_show (std::string prefix,
	   map<grading, basedvector<unsigned, 1> > hq_gens,
	   std::string name,
	   grading delta,
	   mod_map<Z2> f)
{
  printf ("%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
	    printf (", ");
	  printf ("(%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
		printf (", ");
	      printf ("(");
	      for (unsigned k = 1; k <= to_gens.size (); k ++)
		{
		  unsigned gk = to_gens[k];
		  
		  if (k > 1)
		    printf (", ");
		  if (f[gj](gk) == 1)
		    printf ("1");
		  else
		    {
		      assert (f[gj](gk) == 0);
		      printf ("0");
		    }
		}
	      printf (")");
	    }
	  printf ("]");
	}
    }
  printf ("}\n");
}

void
sage_show_khsq (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 (prefix, hq_gens, H);
  sage_show (prefix, hq_gens, "sq1", grading (1, 0), sq1);
  sage_show (prefix, hq_gens, "sq2", grading (2, 0), sq2);
  newline ();
}

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 ("L11n449", sq1.domain (), sq1, sq2);
}

int
main ()
{
  test_sage_show ();
  
#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 1
  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
  knot_diagram kd (rolfsen_knot (5, 2));
  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);
  
  writer w ("sqtest.dat");
  write (w, sq1);
  write (w, sq2);
#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);
	}
  
#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<F> st (kd);
  
	mod_map<R> d2 = st.twisted_d2 ();
	assert (d2.compose (d2) == 0);
	
	mod_map<R> d2U1 = st.twisted_d2Un (1);
	// mod_map<R> d2U1 = st.twisted_d2U1_test ();
  
	assert (d2.compose (d2U1) + d2U1.compose (d2) == 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);

	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);
	
	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
}