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main.cpp
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main.cpp
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#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;
assert (is_even (dwidth));
unsigned hwidth = (dwidth / 2) + 1;
return hwidth;
}
basedvector<unsigned, 1> hwidth_knots;
void
load (map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > &knot_kh_sq,
knot_desc desc)
{
char buf[1000];
if (desc.t == knot_desc::TORUS)
{
sprintf (buf, "knot_kh_sq/T.dat");
}
else
{
unsigned j0 = desc.j;
switch (desc.t)
{
case knot_desc::ROLFSEN:
sprintf (buf, "knot_kh_sq/%d_%d.dat", desc.i, j0);
break;
case knot_desc::HTW:
sprintf (buf, "knot_kh_sq/K%d_%d.dat", desc.i, j0);
break;
case knot_desc::MT:
sprintf (buf, "knot_kh_sq/L%d_%d.dat", desc.i, j0);
break;
default: abort ();
}
}
struct stat stat_buf;
if (stat (buf, &stat_buf) != 0)
{
if (errno == ENOENT)
return;
stderror ("stat: %s", buf);
exit (EXIT_FAILURE);
}
printf ("loading %s...\n", buf);
file_reader r (buf, 1);
map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > m (r);
for (map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > >::const_iter i = m; i; i ++)
{
mod_map<Z2> sq1 = i.val ().first;
ptr<const module<Z2> > H = sq1.domain ();
unsigned hwidth = homological_width (H);
// hwidth_knots[hwidth] ++;
#if 0
if (hwidth == 2)
continue;
if (i.key ().t == knot_desc::MT
&& i.key ().diagram ().num_components () == 1)
continue;
#endif
knot_kh_sq.push (i.key (), i.val ());
}
printf ("done.\n");
}
void
load (map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > &knot_kh_sq,
const char *file)
{
char buf[1000];
sprintf (buf, "knot_kh_sq/%s", file);
printf ("loading %s...\n", buf);
struct stat stat_buf;
if (stat (buf, &stat_buf) != 0)
{
if (errno == ENOENT)
{
printf ("%s does not exist.\n", buf);
return;
}
stderror ("stat: %s", buf);
exit (EXIT_FAILURE);
}
file_reader r (buf, 1);
map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > m (r);
for (map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > >::const_iter i = m; i; i ++)
{
#if 0
mod_map<Z2> sq1 = i.val ().first;
ptr<const module<Z2> > H = sq1.domain ();
unsigned hwidth = homological_width (H);
// hwidth_knots[hwidth] ++;
#endif
#if 0
if (hwidth == 2)
continue;
if (i.key ().t == knot_desc::MT
&& i.key ().diagram ().num_components () == 1)
continue;
#endif
knot_kh_sq.push (i.key (), i.val ());
}
printf ("done.\n");
}
static const int block_size = 100;
void
sage_show (FILE *fp,
std::string prefix,
map<grading, basedvector<unsigned, 1> > hq_gens,
ptr<const module<Z2> > H)
{
fprintf (fp, "%s_hom={", prefix.c_str ());
bool first = 1;
for (map<grading, basedvector<unsigned, 1> >::const_iter i = hq_gens; i; i ++)
{
if (first)
first = 0;
else
fprintf (fp, ", ");
fprintf (fp, "(%d, %d): %d", i.key ().h, i.key ().q, i.val ().size ());
}
fprintf (fp, "}\n");
}
void
sage_show (FILE *fp,
std::string prefix,
map<grading, basedvector<unsigned, 1> > hq_gens,
std::string name,
grading delta,
mod_map<Z2> f)
{
fprintf (fp, "%s_%s={", prefix.c_str (), name.c_str ());
bool first = 1;
for (map<grading, basedvector<unsigned, 1> >::const_iter i = hq_gens; i; i ++)
{
grading from_hq = i.key ();
basedvector<unsigned, 1> from_gens = i.val ();
grading to_hq = from_hq + delta;
if (hq_gens % to_hq)
{
basedvector<unsigned, 1> to_gens = hq_gens(to_hq);
if (first)
first = 0;
else
fprintf (fp, ", ");
fprintf (fp, "(%d, %d): [", from_hq.h, from_hq.q);
bool first2 = 1;
for (unsigned j = 1; j <= from_gens.size (); j ++)
{
unsigned gj = from_gens[j];
if (first2)
first2 = 0;
else
fprintf (fp, ", ");
fprintf (fp, "(");
for (unsigned k = 1; k <= to_gens.size (); k ++)
{
unsigned gk = to_gens[k];
if (k > 1)
fprintf (fp, ", ");
if (f[gj](gk) == 1)
fprintf (fp, "1");
else
{
assert (f[gj](gk) == 0);
fprintf (fp, "0");
}
}
fprintf (fp, ")");
}
fprintf (fp, "]");
}
}
fprintf (fp, "}\n");
}
void
sage_show_khsq (FILE *fp,
std::string prefix,
ptr<const module<Z2> > H,
mod_map<Z2> sq1,
mod_map<Z2> sq2)
{
unsigned n = H->dim ();
map<grading, basedvector<unsigned, 1> > hq_gens;
for (unsigned i = 1; i <= H->dim (); i ++)
hq_gens[H->generator_grading (i)].append (i);
unsigned t = 0;
for (map<grading, basedvector<unsigned, 1> >::const_iter i = hq_gens; i; i ++)
t += i.val ().size ();
assert (t == n);
sage_show (fp, prefix, hq_gens, H);
sage_show (fp, prefix, hq_gens, "sq1", grading (1, 0), sq1);
sage_show (fp, prefix, hq_gens, "sq2", grading (2, 0), sq2);
fprintf (fp, "\n");
}
void
test_sage_show ()
{
knot_diagram kd (mt_link (11, 0, 449));
show (kd); newline ();
planar_diagram pd (kd);
pd.display_knottheory ();
cube<Z2> c (kd);
mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
chain_complex_simplifier<Z2> s (c.khC, d, 1);
steenrod_square sq (c, d, s);
mod_map<Z2> sq1 = sq.sq1 ();
mod_map<Z2> sq2 = sq.sq2 ();
// display ("sq2:\n", sq2);
sage_show_khsq (stdout, "L11n449", sq1.domain (), sq1, sq2);
}
void
dump_sage ()
{
char buf[1000];
#if 0
for (unsigned i = 10; i >= 1; i --)
{
if (i <= 10)
{
for (unsigned j = 1; j <= rolfsen_crossing_knots (i); j += block_size)
{
load (knot_kh_sq, knot_desc (knot_desc::ROLFSEN, i, j));
}
}
...
}
#endif
for (unsigned i = 12; i >= 1; i --)
{
map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > knot_kh_sq;
#if 0
sprintf (buf, "K%d.sage", i);
FILE *fp = fopen (buf, "w");
if (fp == 0)
{
stderror ("fopen: %s", buf);
exit (EXIT_FAILURE);
}
for (unsigned j = 1; j <= htw_knots (i); j += block_size)
{
load (knot_kh_sq, knot_desc (knot_desc::HTW, i, j));
}
for (map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > >::const_iter k = knot_kh_sq; k; k ++)
{
sprintf (buf, "K%s", k.key ().name ().c_str ());
sage_show_khsq (fp, buf, k.val ().first.domain (), k.val ().first, k.val ().second);
}
fclose (fp);
#endif
#if 1
sprintf (buf, "L%d.sage", i);
FILE *fp = fopen (buf, "w");
if (fp == 0)
{
stderror ("fopen: %s", buf);
exit (EXIT_FAILURE);
}
if (i <= 13)
{
for (unsigned j = 1; j <= mt_links (i); j += block_size)
{
load (knot_kh_sq, knot_desc (knot_desc::MT, i, j));
}
}
if (i == 14)
{
for (unsigned j = 1; j <= mt_links (14, 0); j += block_size)
{
load (knot_kh_sq, knot_desc (knot_desc::MT, 14, mt_links (14, 1) + j));
}
}
for (map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > >::const_iter k = knot_kh_sq; k; k ++)
{
sprintf (buf, "K%s", k.key ().name ().c_str ());
sage_show_khsq (fp, buf, k.val ().first.domain (), k.val ().first, k.val ().second);
}
fclose (fp);
#endif
}
}
void
convert_knot_sq ()
{
FILE *fp = open_file ("L14.files", "r");
char buf[1000];
gzfile_writer w ("L14_sq.dat.gz");
w.write_unsigned (76516);
set<knot_desc> keys;
while (fgets (buf, 1000, fp))
{
char *p = strchr (buf, '\n');
if (p)
{
assert (p[1] == 0);
*p = 0;
}
map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > knot_sq;
load (knot_sq, buf);
for (map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > >::const_iter k = knot_sq; k; k ++)
{
write (w, k.key ());
write (w, k.val ());
}
}
// write (w, knot_sq);
}
#if 0
void
test_knot_sq ()
{
const char *knot_sq_files[] = {
"knot_sq/rolfsen_sq.dat.gz",
"knot_sq/Ksmall_sq.dat.gz",
"knot_sq/K11_sq.dat.gz",
"knot_sq/K12_sq.dat.gz",
"knot_sq/Lsmall_sq.dat.gz",
"knot_sq/L11_sq.dat.gz",
"knot_sq/L12_sq.dat.gz",
0,
};
for (unsigned i = 0; knot_sq_files[i] != 0; i ++)
{
gzfile_reader r (knot_sq_files[i]);
map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > knot_sq (r);
printf ("|knot_sq| = %d\n", knot_sq.card ());
}
}
pair<mod_map<Z2>, mod_map<Z2> >
compute_kh_sq (const knot_desc &desc)
{
abort ();
}
#endif
template<class R> pair<ptr<const module<R> >, mod_map<R> >
FU_complex (ptr<const module<R> > CF,
mod_map<R> dF,
mod_map<R> dU)
{
unsigned n = CF->dim ();
basedvector<grading, 1> hq (n * 2);
for (unsigned i = 1; i <= n; i ++)
hq[i] = hq[n + i] = CF->generator_grading (i);
ptr<const module<R> > CFU = new explicit_module<R> (2 * n,
basedvector<R, 1> (),
hq);
map_builder<R> b (CFU);
for (unsigned i = 1; i <= n; i ++)
{
for (linear_combination_const_iter<R> j = dF[i]; j; j ++)
{
b[i].muladd (j.val (), j.key ());
b[n + i].muladd (j.val (), n + j.key ());
}
for (linear_combination_const_iter<R> j = dU[i]; j; j ++)
{
b[i].muladd (j.val (), n + j.key ());
}
}
mod_map<R> dFU (b);
// assert (dFU.compose (dFU) == 0);
// dFU.check_sseq_graded ();
return pair<ptr<const module<R> >, mod_map<R> > (CFU, dFU);
}
void
compute_twistedU ()
{
#if 0
for (unsigned i = 1; i <= 11; i ++)
for (unsigned j = 1; j <= htw_knots (i, 0); j ++)
{
knot_diagram kd (htw_knot (i, 0, j));
...}
#endif
for (unsigned i = 1; i <= 10; i ++)
for (unsigned j = 1; j <= mt_links (i, 0); j ++)
{
knot_diagram kd (mt_link (i, 0, j));
kd.marked_edge = 1;
show (kd); newline ();
typedef spanning_tree_complex<Z2>::R R;
spanning_tree_complex<Z2> c (kd);
ptr<const module<R> > C = c.C;
mod_map<R> d2 = c.twisted_d2 ();
assert (d2.compose (d2) == 0);
mod_map<R> d2U = c.twisted_d2Un (1);
assert (d2.compose (d2U) == d2U.compose (d2));
chain_complex_simplifier<R> s (C, d2, 2);
assert (s.new_d == 0);
printf ("|s.new_C| = %d\n", s.new_C->dim ());
pair<ptr<const module<R> >, mod_map<R> > p = FU_complex (C, d2, d2U);
ptr<const module<R> > CFU = p.first;
mod_map<R> dFU = p.second;
chain_complex_simplifier<R> s2 (CFU, dFU, 2);
assert (s2.new_d == 0);
printf ("|s2.new_C| = %d\n", s2.new_C->dim ());
if (s2.new_C->dim () < 2 * s.new_C->dim ())
printf (" > EXAMPLE\n");
}
}
void
test_forgetful_ss ()
{
typedef fraction_field<polynomial<Z2> > R;
for (unsigned i = 1; i <= 12; i ++)
for (unsigned j = 1; j <= mt_links (i, 0); j ++)
{
knot_diagram kd (mt_link (i, 0, j));
unsigned n = kd.num_components ();
if (n < 2)
continue;
show (kd); newline ();
unionfind<1> u (kd.num_edges ());
for (unsigned i = 1; i <= kd.n_crossings; i ++)
{
u.join (kd.ept_edge (kd.crossings[i][1]),
kd.ept_edge (kd.crossings[i][3]));
u.join (kd.ept_edge (kd.crossings[i][2]),
kd.ept_edge (kd.crossings[i][4]));
}
printf ("%d components:\n", n);
map<unsigned, unsigned> root_comp;
unsigned t = 0;
for (unsigned i = 1; i <= kd.num_edges (); i ++)
{
if (u.find (i) == i)
{
++ t;
root_comp.push (i, t);
}
}
assert (t == n);
printf (" kd w: %d\n", kd.writhe ());
multivariate_laurentpoly<Z> disj_P = 1;
for (unsigned k = 1; k <= n; k ++)
{
knot_diagram comp (SUBLINK, smallbitset (n, unsigned_2pow (k - 1)), kd);
unsigned w = 0;
for (unsigned i = 1; i <= kd.n_crossings; i ++)
{
if (root_comp(u.find (kd.ept_edge (kd.crossings[i][1]))) == k
&& root_comp(u.find (kd.ept_edge (kd.crossings[i][2]))) == k)
{
if (kd.is_to_ept (kd.crossings[i][1]) == kd.is_to_ept (kd.crossings[i][4]))
w ++;
else
w --;
}
}
printf (" % 2d w: %d\n", k, w);
cube<R> c (comp);
mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
chain_complex_simplifier<R> s (c.khC, d, 1);
assert (s.new_d == 0);
multivariate_laurentpoly<Z> P = s.new_C->free_ell_poincare_polynomial ();
printf (" % 2d P: ", k);
display (P);
disj_P *= (P
* multivariate_laurentpoly<Z> (1, VARIABLE, 1, w)
);
}
cube<R> c (kd);
#if 0
for (unsigned i = 0; i < c.n_resolutions; i ++)
{
smallbitset state (kd.n_crossings, i);
smoothing s (kd, state);
s.show_self (kd, state);
newline ();
}
#endif
mod_map<R> untwisted_d = c.compute_d (1, 0, 0, 0, 0);
assert (untwisted_d.compose (untwisted_d) == 0);
chain_complex_simplifier<R> s1 (c.khC, untwisted_d, 1);
assert (s1.new_d == 0);
multivariate_laurentpoly<Z> P1 = s1.new_C->free_ell_poincare_polynomial ();
display (" link P : ", P1);
display (" disj_P (adj): ", disj_P);
mod_map<R> d = untwisted_d;
for (unsigned x = 1; x <= kd.n_crossings; x ++)
{
unsigned r1 = u.find (kd.ept_edge (kd.crossings[x][1])),
r2 = u.find (kd.ept_edge (kd.crossings[x][2]));
d = d + c.compute_dinv (x)*(R (polynomial<Z2> (1, root_comp(r1))) + 1);
d = d + c.compute_dinv (x)*(R (polynomial<Z2> (1, root_comp(r2))) + 1);
#if 0
R hbar (polynomial<Z2> (1, 1));
R hbarp1 = hbar + 1;
// if (u.find (kd.ept_edge (kd.crossings[x][1])) != u.find (kd.ept_edge (kd.crossings[x][2])))
if (u.find (kd.ept_edge (kd.crossings[x][1])) != comp1_root)
d = d + c.compute_dinv (x)*hbarp1;
if (u.find (kd.ept_edge (kd.crossings[x][2])) != comp1_root)
d = d + c.compute_dinv (x)*hbarp1;
#endif
}
#if 0
mod_map<R> h = d.compose (d);
display ("h:\n", h);
#endif
assert (d.compose (d) == 0);
// display ("d:\n", d);
chain_complex_simplifier<R> s2 (c.khC, d, -1);
assert (s2.new_d == 0);
multivariate_laurentpoly<Z> P2 = (s2.new_C->free_ell_poincare_polynomial ()
* multivariate_laurentpoly<Z> (1, VARIABLE, 1, kd.writhe ())
);
display (" Einf P (adj): ", P2);
if (disj_P == P2)
printf (" disj_P == Einf P (adj): YES!\n");
else
printf (" disj_P != Einf P (adj): NO :-(!\n");
fflush (stdout);
}
}
void
compute_forgetful_torsion ()
{
typedef polynomial<Zp<2> > R;
for (unsigned i = 1; i <= 12; i ++)
for (unsigned j = 1; j <= mt_links (i, 0); j ++)
{
knot_diagram kd (mt_link (i, 0, j));
unsigned n = kd.num_components ();
if (n != 2)
continue;
show (kd); newline ();
unionfind<1> u (kd.num_edges ());
for (unsigned i = 1; i <= kd.n_crossings; i ++)
{
u.join (kd.ept_edge (kd.crossings[i][1]),
kd.ept_edge (kd.crossings[i][3]));
u.join (kd.ept_edge (kd.crossings[i][2]),
kd.ept_edge (kd.crossings[i][4]));
}
map<unsigned, unsigned> root_comp;
unsigned t = 0;
for (unsigned i = 1; i <= kd.num_edges (); i ++)
{
if (u.find (i) == i)
{
++ t;
root_comp.push (i, t);
}
}
assert (t == n);
cube<R> c (kd);
#if 0
for (unsigned i = 0; i < c.n_resolutions; i ++)
{
smallbitset state (kd.n_crossings, i);
smoothing s (kd, state);
s.show_self (kd, state);
newline ();
}
#endif
mod_map<R> untwisted_d = c.compute_d (1, 0, 0, 0, 0);
assert (untwisted_d.compose (untwisted_d) == 0);
mod_map<R> d = untwisted_d;
for (unsigned x = 1; x <= kd.n_crossings; x ++)
{
unsigned r1 = u.find (kd.ept_edge (kd.crossings[x][1])),
r2 = u.find (kd.ept_edge (kd.crossings[x][2]));
if (r1 != r2)
d = d + c.compute_dinv (x)*(R (1, 1));
}
#if 0
mod_map<R> h = d.compose (d);
display ("h:\n", h);
#endif
assert (d.compose (d) == 0);
// display ("d:\n", d);
chain_complex_simplifier<R> s (c.khC, d, 0);
printf ("|s.new_C| = %d\n", s.new_C->dim ());
display ("s.new_d:\n", s.new_d);
#if 0
ptr<const free_submodule<R> > ker = s.new_d.kernel ();
printf ("|ker| = %d\n", ker->dim ());
ptr<const free_submodule<R> > im = s.new_d.image ();
printf ("|im| = %d\n", im->dim ());
#endif
ptr<const module<R> > H = s.new_d.homology ();
display ("H:\n", *H);
// printf ("dim H = %d, free rank H = %d\n", H->dim (), H->free_rank ());
}
}
void
test_forgetful_signs ()
{
typedef Q R;
#if 1
for (unsigned i = 1; i <= 14; i ++)
for (unsigned j = 1; j <= mt_links (i, 1); j ++)
{
knot_diagram kd (mt_link (i, 1, j));
#endif
#if 0
for (unsigned i = 2; i <= 16; i ++)
for (unsigned j = i; j <= 16; j ++)
{
knot_diagram kd (torus_knot (i, j));
assert (kd.n_crossings == (i - 1) * j
&& kd.n_crossings <= i * (j - 1));
if (kd.n_crossings > 13)
continue;
}
#endif
unsigned n = kd.num_components ();
if (n < 2)
continue;
assert (n < 97);
show (kd); newline ();
unionfind<1> u (kd.num_edges ());
for (unsigned i = 1; i <= kd.n_crossings; i ++)
{
u.join (kd.ept_edge (kd.crossings[i][1]),
kd.ept_edge (kd.crossings[i][3]));
u.join (kd.ept_edge (kd.crossings[i][2]),
kd.ept_edge (kd.crossings[i][4]));
}
map<unsigned, unsigned> root_comp;
unsigned t = 0;
for (unsigned i = 1; i <= kd.num_edges (); i ++)
{
if (u.find (i) == i)
{
++ t;
root_comp.push (i, t);
}
}
assert (t == n);
basedvector<R, 1> comp_weight (n);
for (unsigned i = 1; i <= n; i ++)
comp_weight[i] = R ((int)i);
map<unsigned, R> crossing_over_sign;
// crossings
set<unsigned> pending;
set<unsigned> finished;
crossing_over_sign.push (1, R (1));