Ready to test forgetful spectral sequence on links through 11

crossings.
This commit is contained in:
Cotton Seed 2012-08-30 18:30:39 -04:00
parent e9f41377ee
commit 6e46c5a8ce
5 changed files with 237 additions and 2 deletions

View File

@ -942,6 +942,7 @@ class mod_map
// ?? add and other map operations should not be explicit // ?? add and other map operations should not be explicit
mod_map operator + (const mod_map &m) const; mod_map operator + (const mod_map &m) const;
mod_map operator * (const R &c) const;
bool homogeneous () const; bool homogeneous () const;
void check_grading (grading delta) const; void check_grading (grading delta) const;
@ -1787,6 +1788,14 @@ mod_map<R>::check_grading (grading delta) const
} }
} }
template<class R> mod_map<R>
mod_map<R>::operator * (const R &c) const
{
basedvector<linear_combination<R>, 1> v (impl->from->dim ());
for (unsigned i = 1; i <= impl->from->dim (); i ++)
v[i] = c*column (i);
return mod_map (IMPL, new explicit_map_impl<R> (impl->from, impl->to, v));
}
template<class R> mod_map<R> template<class R> mod_map<R>
mod_map<R>::operator + (const mod_map &m) const mod_map<R>::operator + (const mod_map &m) const

1
cube.h
View File

@ -47,6 +47,7 @@ public:
bool reverse_orientation, bool reverse_orientation,
unsigned to_reverse) const; unsigned to_reverse) const;
mod_map<R> compute_dinv (unsigned c);
mod_map<R> H_i (unsigned c); mod_map<R> H_i (unsigned c);
mod_map<R> compute_nu () const; mod_map<R> compute_nu () const;

View File

@ -356,6 +356,96 @@ cube<R>::H_i (unsigned c)
return H_c; return H_c;
} }
template<class R> mod_map<R>
cube<R>::compute_dinv (unsigned c)
{
map_builder<R> p (khC);
for (unsigned i = 0; i < n_resolutions; i ++)
{
if (!unsigned_bittest (i, c))
continue;
smoothing from_s (kd, smallbitset (n_crossings, i));
unsigned i2 = unsigned_bitclear (i, c);
smoothing to_s (kd, smallbitset (n_crossings, i2));
basedvector<unsigned, 1> from_circle_edge_rep (from_s.n_circles);
for (unsigned j = 1; j <= kd.num_edges (); j ++)
from_circle_edge_rep[from_s.edge_circle[j]] = j;
unsigned a = from_s.crossing_from_circle (kd, c),
b = from_s.crossing_to_circle (kd, c);
unsigned x = to_s.crossing_from_circle (kd, c),
y = to_s.crossing_to_circle (kd, c);
for (unsigned j = 0; j < from_s.num_monomials (); j ++)
{
unsigned j2 = 0;
for (unsigned_const_iter k = j; k; k ++)
{
unsigned s = k.val ();
j2 = unsigned_bitset (j2, to_s.edge_circle[from_circle_edge_rep[s]]);
}
if (a == b)
{
// split
assert (x != y);
if (unsigned_bittest (j, a))
{
// 1 -> x + y
j2 = unsigned_bitset (j2, x);
j2 = unsigned_bitclear (j2, y);
p[generator (i, j)].muladd (1, generator (i2, j2));
j2 = unsigned_bitclear (j2, x);
j2 = unsigned_bitset (j2, y);
p[generator (i, j)].muladd (1, generator (i2, j2));
}
else
{
// a -> xy
j2 = unsigned_bitclear (j2, x);
j2 = unsigned_bitclear (j2, y);
p[generator (i, j)].muladd (1, generator (i2, j2));
}
}
else
{
// join
assert (x == y);
if (unsigned_bittest (j, a)
&& unsigned_bittest (j, b))
{
// 1 -> 1
j2 = unsigned_bitset (j2, x);
p[generator (i, j)].muladd (1, generator (i2, j2));
}
else if ((unsigned_bittest (j, a)
&& !unsigned_bittest (j, b))
|| (!unsigned_bittest (j, a)
&& unsigned_bittest (j, b)))
{
// a, b -> x
j2 = unsigned_bitclear (j2, x);
p[generator (i, j)].muladd (1, generator (i2, j2));
}
}
}
}
mod_map<R> dinv (p);
dinv.check_grading (grading (-1, -2));
return dinv;
}
template<class R> void template<class R> void
cube<R>::check_reverse_crossings () cube<R>::check_reverse_crossings ()
{ {

134
main.cpp
View File

@ -777,9 +777,143 @@ compute_twistedU ()
} }
} }
void
test_forgetful_ss ()
{
typedef fraction_field<polynomial<Z2> > R;
for (unsigned i = 1; i <= 11; 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);
unsigned disj_rank = 1;
for (unsigned k = 1; k <= n; k ++)
{
knot_diagram comp (SUBLINK, smallbitset (n, unsigned_2pow (k - 1)), kd);
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);
printf (" % 2d: rank %d\n", k, s.new_C->dim ());
disj_rank *= s.new_C->dim ();
}
{
knot_diagram comp (SUBLINK, smallbitset (n, unsigned_bitclear (unsigned_fill (n), 1)), kd);
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);
printf (" 11...10: rank %d\n", s.new_C->dim ());
}
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);
printf ("untwisted rank = %d\n", s1.new_C->dim ());
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);
printf ("twisted rank = %d\n", s2.new_C->dim ());
if (disj_rank == s2.new_C->dim ())
printf (" %d == %d: YES!\n", disj_rank, s2.new_C->dim ());
else
printf (" %d == %d: NO :-(\n", disj_rank, s2.new_C->dim ());
}
}
int int
main () main ()
{ {
test_forgetful_ss ();
return 0;
compute_twistedU (); compute_twistedU ();
#if 1 #if 1

View File

@ -172,9 +172,10 @@ 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 ++) for (linear_combination_const_iter<R> j = new_d_columns[i]; j; j ++)
{ {
grading jgr = C->generator_grading (j.key ()); grading jgr = C->generator_grading (j.key ());
assert (jgr.h >= igr.h); // assert (jgr.h >= igr.h);
if (j.val ().is_unit () if (j.val ().is_unit ()
&& jgr.h - igr.h == dh) // && jgr.h - igr.h == dh
)
{ {
cancel (i, j.val (), j.key ()); cancel (i, j.val (), j.key ());
break; break;