Added tests for Kh homotopy type: CP^2, mutation invariance, sq1 and

sq2 satisfy adem relations.
This commit is contained in:
Cotton Seed 2012-09-09 11:18:25 -04:00
parent 307719de67
commit b17121cf5b
2 changed files with 444 additions and 79 deletions

View File

@ -86,6 +86,13 @@ class knot_desc
std::string name () const; std::string name () const;
unsigned table_crossing_knots () const; unsigned table_crossing_knots () const;
hash_t hash_self () const
{
return hash_combine (hash ((int)t),
hash_combine (hash (i),
hash (j)));
}
void write_self (writer &w) const; void write_self (writer &w) const;
void show_self () const { printf ("%s", name ().c_str ()); } void show_self () const { printf ("%s", name ().c_str ()); }
void display_self () const { show_self (); newline (); } void display_self () const { show_self (); newline (); }

516
main.cpp
View File

@ -219,6 +219,7 @@ homological_width (ptr<const module<Z2> > H)
} }
int dwidth = maxd - mind; int dwidth = maxd - mind;
assert (is_even (dwidth));
unsigned hwidth = (dwidth / 2) + 1; unsigned hwidth = (dwidth / 2) + 1;
return hwidth; return hwidth;
} }
@ -346,84 +347,6 @@ load (map<knot_desc,
static const int block_size = 100; 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 void
sage_show (FILE *fp, sage_show (FILE *fp,
std::string prefix, std::string prefix,
@ -669,6 +592,7 @@ convert_knot_sq ()
// write (w, knot_sq); // write (w, knot_sq);
} }
#if 0
void void
test_knot_sq () test_knot_sq ()
{ {
@ -697,6 +621,7 @@ compute_kh_sq (const knot_desc &desc)
{ {
abort (); abort ();
} }
#endif
template<class R> pair<ptr<const module<R> >, mod_map<R> > template<class R> pair<ptr<const module<R> >, mod_map<R> >
FU_complex (ptr<const module<R> > CF, FU_complex (ptr<const module<R> > CF,
@ -923,12 +848,445 @@ test_forgetful_ss ()
} }
} }
/* Kh homotopy type:
- check for CP^2,
- mutation invariance
- K-theory */
map<grading, basedvector<int, 1> >
compute_st (mod_map<Z2> sq1,
mod_map<Z2> sq2)
{
ptr<const module<Z2> > H = sq1.domain ();
map<grading, basedvector<int, 1> > st;
bool first = 1;
set<grading> gs = H->gradings ();
for (set_const_iter<grading> gg = gs; gg; gg ++)
{
grading hq = gg.val (),
h1q (hq.h + 1, hq.q),
h2q (hq.h + 2, hq.q);
ptr<const free_submodule<Z2> > H_hq = H->graded_piece (hq),
H_h1q = H->graded_piece (h1q),
H_h2q = H->graded_piece (h2q);
mod_map<Z2> S = sq2.restrict (H_hq, H_h2q),
A = sq1.restrict (H_hq, H_h1q),
B = sq1.restrict (H_h1q, H_h2q);
ptr<const free_submodule<Z2> > S_im = S.image (),
A_ker = A.kernel (),
B_im = B.image ();
ptr<const free_submodule<Z2> > inter = S_im->intersection (B_im);
mod_map<Z2> S_res = S.restrict_from (A_ker);
ptr<const free_submodule<Z2> > S_res_im = S_res.image ();
ptr<const free_submodule<Z2> > res_inter = S_res_im->intersection (B_im);
int r1 = S_im->dim ();
int r2 = S_res_im->dim ();
int r3 = inter->dim ();
int r4 = res_inter->dim ();
int s1 = r2 - r4,
s2 = r1 - r2 - r3 + r4,
s3 = r4,
s4 = r3 - r4;
if (!s1 && !s2 && !s3 && !s4)
continue;
basedvector<int, 1> s (4);
s[1] = s1;
s[2] = s2;
s[3] = s3;
s[4] = s4;
st.push (hq, s);
}
return st;
}
void
test_knot_sq ()
{
const char *knot_sq_files[] = {
"knot_sq/rolfsen_sq.dat.gz",
"knot_sq/Ksmall_sq.dat.gz",
"knot_sq/K11_sq.dat.gz",
"knot_sq/K12_sq.dat.gz",
"knot_sq/K13_sq.dat.gz",
#if 0
"knot_sq/K14_sq.dat.gz",
"knot_sq/Lsmall_sq.dat.gz",
"knot_sq/L11_sq.dat.gz",
"knot_sq/L12_sq.dat.gz",
"knot_sq/L13_sq.dat.gz",
"knot_sq/L14_sq.dat.gz",
#endif
0,
};
hashset<knot_desc> mutants;
for (unsigned i = 11; i <= 15; i ++)
{
basedvector<basedvector<unsigned, 1>, 1> groups
= mutant_knot_groups (i);
for (unsigned j = 1; j <= groups.size (); j ++)
{
for (unsigned k = 1; k <= groups[j].size (); k ++)
mutants += knot_desc (knot_desc::HTW, i, groups[j][k]);
}
}
printf ("|mutants| = %d\n", mutants.card ());
basedvector<unsigned, 1> width_hist (7);
for (unsigned i = 1; i <= width_hist.size (); i ++)
width_hist[i] = 0;
hashmap<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > mutant_knot_sq,
mutant_mknot_sq;
{
gzfile_reader r ("knot_sq/mut_mknot.dat.gz");
mutant_mknot_sq = hashmap<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > (r);
for (hashmap<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > >::const_iter i = mutant_mknot_sq; i; i ++)
{
knot_desc desc = i.key ();
// careful: duplicate code!
pair<mod_map<Z2>, mod_map<Z2> > p = i.val ();
mod_map<Z2> sq1 = p.first,
sq2 = p.second;
ptr<const module<Z2> > H = sq1.domain ();
unsigned w = homological_width (H);
assert (w >= 1 && w <= 7);
width_hist[w] ++;
assert (sq1.compose (sq1) == 0);
assert (sq2.compose (sq2) == sq1.compose (sq2).compose (sq1));
map<grading, basedvector<int, 1> > st
= compute_st (sq1, sq2);
for (map_const_iter<grading, basedvector<int, 1> > i = st; i; i ++)
{
if (i.val ()[1] != 0)
{
show (desc); newline ();
printf (" > has CP^2\n");
}
}
}
}
for (unsigned i = 0; knot_sq_files[i]; i ++)
{
gzfile_reader r (knot_sq_files[i]);
printf ("loading %s...\n", knot_sq_files[i]);
unsigned n = r.read_unsigned ();
for (unsigned i = 1; i <= n; i ++)
{
knot_desc desc (r);
pair<mod_map<Z2>, mod_map<Z2> > p (r);
if (mutants % desc)
mutant_knot_sq.push (desc, p);
mod_map<Z2> sq1 = p.first,
sq2 = p.second;
ptr<const module<Z2> > H = sq1.domain ();
unsigned w = homological_width (H);
assert (w >= 1 && w <= 7);
width_hist[w] ++;
assert (sq1.compose (sq1) == 0);
assert (sq2.compose (sq2) == sq1.compose (sq2).compose (sq1));
map<grading, basedvector<int, 1> > st
= compute_st (sq1, sq2);
for (map_const_iter<grading, basedvector<int, 1> > i = st; i; i ++)
{
if (i.val ()[1] != 0)
{
show (desc); newline ();
printf (" > has CP^2\n");
}
}
}
}
printf ("|mutant_knot_sq| = %d\n", mutant_knot_sq.card ());
printf ("width_hist:\n");
for (unsigned i = 1; i <= width_hist.size (); i ++)
{
printf (" % 2d: %d\n", i, width_hist[i]);
}
unsigned missing = 0,
compared = 0;
for (unsigned i = 11; i <= 15; i ++)
{
basedvector<basedvector<unsigned, 1>, 1> groups
= mutant_knot_groups (i);
for (unsigned j = 1; j <= groups.size (); j ++)
{
bool first = 1;
knot_desc desc_1;
ptr<const module<Z2> > H_1;
map<grading, basedvector<int, 1> > st_1;
for (unsigned k = 1; k <= groups[j].size (); k ++)
{
knot_desc desc (knot_desc::HTW, i, groups[j][k]);
if (mutant_knot_sq % desc)
{
pair<mod_map<Z2>, mod_map<Z2> > p = mutant_knot_sq(desc);
mod_map<Z2> sq1 = p.first,
sq2 = p.second;
ptr<const module<Z2> > H = sq1.domain ();
map<grading, basedvector<int, 1> > st = compute_st (sq1, sq2);
if (first)
{
first = 0;
desc_1 = desc;
H_1 = H;
st_1 = st;
}
else
{
if (H_1->free_poincare_polynomial ()
== H->free_poincare_polynomial ())
{
if (st_1 != st)
{
printf ("> DIFFER\n");
display (" ", desc_1);
display (" ", desc);
}
compared ++;
}
else if (mutant_mknot_sq % desc)
{
pair<mod_map<Z2>, mod_map<Z2> > mp = mutant_mknot_sq(desc);
mod_map<Z2> msq1 = mp.first,
msq2 = mp.second;
ptr<const module<Z2> > mH = msq1.domain ();
assert (H_1->free_poincare_polynomial ()
== mH->free_poincare_polynomial ());
map<grading, basedvector<int, 1> > mst
= compute_st (msq1, msq2);
if (st_1 != mst)
{
printf ("> DIFFER\n");
display (" ", desc_1);
display (" ", desc);
}
compared ++;
}
else
missing ++;
}
}
else
missing ++;
}
}
}
printf ("missing = %d\n", missing);
printf ("compared = %d\n", compared);
}
void
compute_mutant_mirrors ()
{
const char *knot_sq_files[] = {
// "knot_sq/K11_sq.dat.gz",
// "knot_sq/K12_sq.dat.gz",
// "knot_sq/K13_sq.dat.gz",
"knot_sq/K14_sq.dat.gz",
0,
};
unsigned minn = 14,
maxn = 14;
hashset<knot_desc> mutants;
for (unsigned i = minn; i <= maxn; i ++)
{
basedvector<basedvector<unsigned, 1>, 1> groups
= mutant_knot_groups (i);
for (unsigned j = 1; j <= groups.size (); j ++)
{
for (unsigned k = 1; k <= groups[j].size (); k ++)
mutants += knot_desc (knot_desc::HTW, i, groups[j][k]);
}
}
printf ("|mutants| = %d\n", mutants.card ());
hashmap<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > mutant_knot_sq;
for (unsigned i = 0; knot_sq_files[i]; i ++)
{
gzfile_reader r (knot_sq_files[i]);
printf ("loading %s...\n", knot_sq_files[i]);
unsigned n = r.read_unsigned ();
for (unsigned i = 1; i <= n; i ++)
{
knot_desc k (r);
pair<mod_map<Z2>, mod_map<Z2> > p (r);
if (mutants % k)
mutant_knot_sq.push (k, p);
}
}
printf ("|mutant_knot_sq| = %d\n", mutant_knot_sq.card ());
map<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > mknot_kh_sq;
for (unsigned i = minn; i <= maxn; i ++)
{
basedvector<basedvector<unsigned, 1>, 1> groups
= mutant_knot_groups (i);
for (unsigned j = 1; j <= groups.size (); j ++)
{
unsigned n = groups[j].size ();
basedvector<ptr<const module<Z2> >, 1> group_H (n);
for (unsigned k = 1; k <= n; k ++)
{
knot_desc desc (knot_desc::HTW, i, groups[j][k]);
pair<mod_map<Z2>, mod_map<Z2> > p = mutant_knot_sq(desc);
group_H[k] = p.first.domain ();
}
for (unsigned k = 2; k <= n; k ++)
{
if (group_H[1]->free_poincare_polynomial ()
!= group_H[k]->free_poincare_polynomial ())
{
knot_desc desc (knot_desc::HTW, i, groups[j][k]);
show (desc); newline ();
knot_diagram kd (MIRROR, desc.diagram ());
cube<Z2> c (kd);
mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
chain_complex_simplifier<Z2> s (c.khC, d, 1);
assert (s.new_d == 0);
steenrod_square sq (c, d, s);
mod_map<Z2> sq1 = sq.sq1 ();
mod_map<Z2> sq2 = sq.sq2 ();
ptr<const module<Z2> > H = sq1.domain ();
assert (group_H[1]->free_poincare_polynomial ()
== H->free_poincare_polynomial ());
char buf[1000];
assert (desc.t == knot_desc::HTW);
sprintf (buf, "mknot/K%d_%d.dat.gz", desc.i, desc.j);
gzfile_writer w (buf);
write (w, desc);
write (w, sq1);
write (w, sq2);
}
}
}
}
}
void
convert_mknots ()
{
hashmap<knot_desc,
pair<mod_map<Z2>, mod_map<Z2> > > mutant_mknot_sq;
for (unsigned i = 11; i <= 15; i ++)
{
basedvector<basedvector<unsigned, 1>, 1> groups
= mutant_knot_groups (i);
for (unsigned j = 1; j <= groups.size (); j ++)
{
unsigned n = groups[j].size ();
for (unsigned k = 1; k <= n; k ++)
{
knot_desc desc (knot_desc::HTW, i, groups[j][k]);
char buf[1000];
assert (desc.t == knot_desc::HTW);
sprintf (buf, "mknot/K%d_%d.dat.gz", desc.i, desc.j);
struct stat stat_buf;
if (stat (buf, &stat_buf) != 0)
{
if (errno != ENOENT)
{
stderror ("stat: %s", buf);
exit (EXIT_FAILURE);
}
}
else
{
printf ("loading %s...\n", buf);
gzfile_reader r (buf);
knot_desc desc_2 (r);
assert (desc_2 == desc);
mod_map<Z2> sq1 (r);
mod_map<Z2> sq2 (r);
mutant_mknot_sq.push (desc,
pair<mod_map<Z2>, mod_map<Z2> > (sq1, sq2));
}
}
}
}
gzfile_writer w ("knot_sq/mut_mknot.dat.gz");
write (w, mutant_mknot_sq);
}
int int
main () main ()
{ {
test_forgetful_ss (); compute_mutant_mirrors ();
return 0; return 0;
test_knot_sq ();
convert_mknots ();
test_forgetful_ss ();
compute_twistedU (); compute_twistedU ();
#if 1 #if 1