Some changes to algebra/ to make kernel, image and quotient work over
PID. Goal: compute spectral sequences over PID. Updated main to compute Lee Z/2 bound.
This commit is contained in:
Cotton Seed
parent
f2375a2d9e
commit
9ee5950195
4
Makefile
4
Makefile
@ -8,8 +8,8 @@ CXX = clang++ -fno-color-diagnostics --stdlib=libc++ --std=c++11 -I/u/cseed/llvm
|
||||
|
||||
INCLUDES = -I/opt/local/include -I.
|
||||
|
||||
# OPTFLAGS = -g
|
||||
OPTFLAGS = -O2 -g
|
||||
OPTFLAGS = -g
|
||||
# OPTFLAGS = -O2 -g
|
||||
# OPTFLAGS = -O2 -DNDEBUG
|
||||
|
||||
LDFLAGS = -L/opt/local/lib -L/u/cseed/llvm-3.1/lib
|
||||
|
||||
@ -41,6 +41,12 @@ class Z2
|
||||
Z2 &operator *= (const Z2 &x) { v &= x.v; return *this; }
|
||||
Z2 &operator /= (const Z2 &x) { assert (x.v); return *this; }
|
||||
|
||||
// *this += z1*z2
|
||||
Z2 &muladdeq (const Z2 &z1, const Z2 &z2)
|
||||
{
|
||||
return operator += (z1 * z2);
|
||||
}
|
||||
|
||||
bool divides (Z2 x) const { return v || !x.v; }
|
||||
bool operator | (const Z2 x) const { return divides (x); }
|
||||
|
||||
|
||||
@ -63,7 +63,12 @@ class module : public refcounted
|
||||
if (i <= free_rank ())
|
||||
return c == 0;
|
||||
else
|
||||
return generator_ann (i) | c;
|
||||
{
|
||||
// ???
|
||||
abort ();
|
||||
|
||||
// return generator_ann (i) | c;
|
||||
}
|
||||
}
|
||||
|
||||
R annihilator (R c, unsigned i) const
|
||||
@ -1853,7 +1858,15 @@ mod_map<R>::kernel () const
|
||||
linear_combination<R> &to_x = to_xs[j],
|
||||
&from_x = from_xs[j];
|
||||
R to_xc = to_x(i);
|
||||
if (! (to_vc | to_xc))
|
||||
if (to_xc == 0)
|
||||
continue;
|
||||
|
||||
if (to_vc == 0 && to_xc != 0)
|
||||
{
|
||||
to_v += to_x;
|
||||
from_v += from_x;
|
||||
}
|
||||
else if (! (to_vc | to_xc))
|
||||
{
|
||||
tuple<R, R, R> t = to_vc.extended_gcd (to_xc);
|
||||
assert (get<0> (t) == to_vc*get<1> (t) + get<2> (t)*to_xc);
|
||||
@ -1959,8 +1972,14 @@ mod_span<R>::mod_span (ptr<const module<R> > mod,
|
||||
|
||||
linear_combination<R> &x = xs[j];
|
||||
R xc = x(i);
|
||||
if (xc == 0)
|
||||
continue;
|
||||
|
||||
if (! (vc | xc))
|
||||
if (vc == 0)
|
||||
{
|
||||
v += x;
|
||||
}
|
||||
else if (! (vc | xc))
|
||||
{
|
||||
tuple<R, R, R> t = vc.extended_gcd (xc);
|
||||
assert (get<0> (t) == vc*get<1> (t) + get<2> (t)*xc);
|
||||
|
||||
@ -219,7 +219,8 @@ class multivariate_laurentpoly
|
||||
|
||||
multivariate_laurentpoly (const multivariate_laurentpoly &p) : coeffs(p.coeffs) { }
|
||||
multivariate_laurentpoly (copy, const multivariate_laurentpoly &p)
|
||||
: coeffs(COPY2, p.coeffs)
|
||||
// ??? COPY2?
|
||||
: coeffs(COPY, p.coeffs)
|
||||
{
|
||||
}
|
||||
|
||||
|
||||
@ -67,7 +67,7 @@ public:
|
||||
|
||||
polynomial recip () const
|
||||
{
|
||||
assert (coeffs.card () != 1);
|
||||
assert (coeffs.card () == 1);
|
||||
|
||||
pair<unsigned, T> p = coeffs.head ();
|
||||
assert (p.first == 0);
|
||||
@ -134,13 +134,16 @@ public:
|
||||
polynomial operator * (const polynomial &p) const;
|
||||
|
||||
bool divides (const polynomial &num) const;
|
||||
|
||||
// *this | num
|
||||
bool operator | (const polynomial &num) const { return divides (num); }
|
||||
|
||||
tuple<polynomial, polynomial> divide_with_remainder (const polynomial &denom) const;
|
||||
|
||||
polynomial mod (const polynomial &denom) const;
|
||||
|
||||
|
||||
polynomial divide_exact (const polynomial &denom) const;
|
||||
polynomial div (const polynomial &denom) const { return divide_exact (denom); }
|
||||
|
||||
polynomial gcd (const polynomial &b) const;
|
||||
polynomial lcm (const polynomial &b) const;
|
||||
@ -392,14 +395,14 @@ class polynomial<Z2>
|
||||
|
||||
public:
|
||||
polynomial () { }
|
||||
explicit polynomial (int x)
|
||||
polynomial (int x)
|
||||
{
|
||||
Z2 c (x);
|
||||
if (c != 0)
|
||||
coeffs.push (0);
|
||||
}
|
||||
|
||||
explicit polynomial (Z2 c)
|
||||
polynomial (Z2 c)
|
||||
{
|
||||
if (c != 0)
|
||||
coeffs.push (0);
|
||||
@ -450,6 +453,17 @@ class polynomial<Z2>
|
||||
bool operator != (int x) const { return !operator == (x); }
|
||||
|
||||
unsigned degree () const { return coeffs.tail (); }
|
||||
bool is_unit () const
|
||||
{
|
||||
return coeffs.card () == 1
|
||||
&& coeffs.head () == 0;
|
||||
}
|
||||
|
||||
polynomial recip () const
|
||||
{
|
||||
assert (is_unit ());
|
||||
return 1;
|
||||
}
|
||||
|
||||
polynomial &operator += (polynomial p) { coeffs ^= p.coeffs; return *this; }
|
||||
polynomial &operator -= (polynomial p) { coeffs ^= p.coeffs; return *this; }
|
||||
|
||||
16
cube_impl.h
16
cube_impl.h
@ -166,7 +166,7 @@ cube<R>::compute_map (unsigned dh, unsigned max_n,
|
||||
continue;
|
||||
assert (!unsigned_bittest (v_to, p));
|
||||
}
|
||||
|
||||
|
||||
b[generator (fromstate, v_from)].muladd
|
||||
(sign, generator (tostate, v_to));
|
||||
}
|
||||
@ -302,7 +302,7 @@ cube<R>::compute_X (unsigned p) const
|
||||
template<class R> mod_map<R>
|
||||
cube<R>::H_i (unsigned c)
|
||||
{
|
||||
mod_map<R> b (khC, 0);
|
||||
map_builder<R> b (khC, 0);
|
||||
for (unsigned i = 0; i < n_resolutions; i ++)
|
||||
{
|
||||
if (unsigned_bittest (i, c))
|
||||
@ -319,6 +319,14 @@ cube<R>::H_i (unsigned c)
|
||||
|
||||
unsigned from = from_s.crossing_from_circle (kd, c),
|
||||
to = from_s.crossing_to_circle (kd, c);
|
||||
|
||||
int sign = 1;
|
||||
for (unsigned j = 1; j < c; j ++)
|
||||
{
|
||||
if (unsigned_bittest (i, j))
|
||||
sign *= -1;
|
||||
}
|
||||
|
||||
for (unsigned j = 0; j < from_s.num_monomials (); j ++)
|
||||
{
|
||||
unsigned j2 = 0;
|
||||
@ -336,7 +344,7 @@ cube<R>::H_i (unsigned c)
|
||||
to_s.crossing_from_circle (kd, c));
|
||||
j2 = unsigned_bitset (j2,
|
||||
to_s.crossing_to_circle (kd, c));
|
||||
v.muladd (1, generator (i2, j2));
|
||||
v.muladd (-sign, generator (i2, j2));
|
||||
}
|
||||
else if (from != to
|
||||
&& !unsigned_bittest (j, from)
|
||||
@ -344,7 +352,7 @@ cube<R>::H_i (unsigned c)
|
||||
{
|
||||
assert (! unsigned_bittest (j2,
|
||||
to_s.crossing_from_circle (kd, c)));
|
||||
v.muladd (1, generator (i2, j2));
|
||||
v.muladd (sign, generator (i2, j2));
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
329
main.cpp
329
main.cpp
@ -850,18 +850,104 @@ test_forgetful_ss ()
|
||||
}
|
||||
|
||||
void
|
||||
test_forgetful_signs ()
|
||||
compute_forgetful_torsion ()
|
||||
{
|
||||
typedef Zp<97> R;
|
||||
typedef polynomial<Zp<2> > R;
|
||||
|
||||
#if 0
|
||||
for (unsigned i = 1; i <= 14; i ++)
|
||||
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 ++)
|
||||
{
|
||||
@ -870,6 +956,7 @@ test_forgetful_signs ()
|
||||
&& kd.n_crossings <= i * (j - 1));
|
||||
if (kd.n_crossings > 13)
|
||||
continue;
|
||||
}
|
||||
#endif
|
||||
|
||||
unsigned n = kd.num_components ();
|
||||
@ -1026,6 +1113,110 @@ test_forgetful_signs ()
|
||||
}
|
||||
}
|
||||
|
||||
void
|
||||
compute_lee_bound ()
|
||||
{
|
||||
typedef Z2 R;
|
||||
|
||||
#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
|
||||
#if 1
|
||||
for (unsigned i = 1; i <= 10; i ++)
|
||||
for (unsigned j = 1; j <= rolfsen_crossing_knots (i); j ++)
|
||||
{
|
||||
knot_diagram kd (rolfsen_knot (i, j));
|
||||
#endif
|
||||
|
||||
show (kd); newline ();
|
||||
|
||||
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> d = c.compute_d (1, 0, 0, 0, 0);
|
||||
assert (d.compose (d) == 0);
|
||||
|
||||
for (unsigned x = 1; x <= kd.n_crossings; x ++)
|
||||
d = d + c.H_i (x);
|
||||
|
||||
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;
|
||||
}
|
||||
}
|
||||
|
||||
printf ("kinf = %d\n", kinf);
|
||||
}
|
||||
}
|
||||
|
||||
void
|
||||
test_crossing_ss ()
|
||||
{
|
||||
typedef multivariate_laurentpoly<Z2> R;
|
||||
|
||||
for (unsigned i = 1; i <= 14; i ++)
|
||||
for (unsigned j = 1; j <= htw_knots (i); j ++)
|
||||
{
|
||||
knot_diagram kd (htw_knot (i, j));
|
||||
show (kd); newline ();
|
||||
|
||||
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 ++)
|
||||
{
|
||||
d = d + c.compute_dinv (x)*(R (1, VARIABLE, x));
|
||||
}
|
||||
|
||||
display ("d:\n", d);
|
||||
|
||||
mod_map<R> h = d.compose (d);
|
||||
display ("h:\n", h);
|
||||
}
|
||||
}
|
||||
|
||||
/* Kh homotopy type:
|
||||
- check for CP^2,
|
||||
- mutation invariance
|
||||
@ -1752,12 +1943,138 @@ convert_15 ()
|
||||
write (w, knot15_sq);
|
||||
}
|
||||
|
||||
void
|
||||
compare_gss_splitting ()
|
||||
{
|
||||
typedef Z2 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
|
||||
|
||||
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 1
|
||||
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
|
||||
|
||||
printf ("inter-component x:\n");
|
||||
mod_map<R> ds (c.khC);
|
||||
for (unsigned x = 1; x <= kd.n_crossings; x ++)
|
||||
{
|
||||
unsigned p1 = kd.crossings[x][1],
|
||||
p2 = kd.crossings[x][2];
|
||||
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)
|
||||
{
|
||||
printf (" %d\n", x);
|
||||
|
||||
ds = ds + c.compute_dinv (x);
|
||||
}
|
||||
}
|
||||
|
||||
mod_map<R> d1 = c.compute_d (1, 0, 0, 0, 0);
|
||||
assert (d1.compose (ds) == ds.compose (d1));
|
||||
|
||||
mod_map<R> d2 = c.compute_d (2, 0, 0, 0, 0);
|
||||
mod_map<R> h2 = d2.compose (ds) + ds.compose (d2);
|
||||
if (h2 != 0)
|
||||
display ("h2:\n", h2);
|
||||
|
||||
mod_map<R> d3 = c.compute_d (3, 0, 0, 0, 0);
|
||||
mod_map<R> h3 = d3.compose (ds) + ds.compose (d3);
|
||||
if (h3 != 0)
|
||||
display ("h3:\n", h3);
|
||||
|
||||
mod_map<R> d = c.compute_d (0, 0, 0, 0, 0);
|
||||
mod_map<R> h = d.compose (ds) + ds.compose (d);
|
||||
if (h != 0)
|
||||
display ("h:\n", h);
|
||||
}
|
||||
}
|
||||
|
||||
int
|
||||
main ()
|
||||
{
|
||||
test_forgetful_signs ();
|
||||
compute_lee_bound ();
|
||||
return 0;
|
||||
|
||||
compare_gss_splitting ();
|
||||
|
||||
{
|
||||
knot_diagram kd (rolfsen_knot (3, 1));
|
||||
cube<Q> c (kd);
|
||||
mod_map<Q> d = c.compute_d (1, 0, 0, 0, 0);
|
||||
chain_complex_simplifier<Q> s (c.khC, d, 0);
|
||||
assert (s.new_d == 0);
|
||||
display ("s.new_C:\n", *s.new_C);
|
||||
}
|
||||
return 0;
|
||||
|
||||
test_forgetful_signs ();
|
||||
|
||||
{
|
||||
knot_diagram kd (torus_knot (2, 2));
|
||||
printf ("n+ = %d, n- = %d\n", kd.nplus, kd.nminus);
|
||||
cube<Z2> c (kd);
|
||||
sseq ss = compute_szabo_sseq (c);
|
||||
ss.texshow (stdout, "Hopf");
|
||||
}
|
||||
{
|
||||
knot_diagram kd (MIRROR, knot_diagram (torus_knot (2, 2)));
|
||||
printf ("n+ = %d, n- = %d\n", kd.nplus, kd.nminus);
|
||||
cube<Z2> c (kd);
|
||||
sseq ss = compute_szabo_sseq (c);
|
||||
ss.texshow (stdout, "Hopf");
|
||||
}
|
||||
|
||||
return 0;
|
||||
|
||||
compute_forgetful_torsion ();
|
||||
return 0;
|
||||
|
||||
test_forgetful_signs ();
|
||||
test_crossing_ss ();
|
||||
|
||||
find_width4_in_h0 ();
|
||||
// convert_15 ();
|
||||
// test_knot_sq ();
|
||||
|
||||
Loading…
Reference in New Issue
Block a user