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First shot at periodicity congruences

This commit is contained in:
Wojciech Politarczyk 2016-11-19 20:53:18 +01:00
parent df8f02f3ca
commit cddb533519
2 changed files with 110 additions and 30 deletions
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52
algebra/multivariate_laurentpoly.h
View File

@ -481,3 +481,55 @@ multivariate_laurentpoly<T>::show_self () const
if (first)
printf ("0");
}
// functions below were added to verify several periodicity criteria
// function below inverts every occurence of a variable x_index
// i.e. this means that x_index is replaced by x_index^{-1}
template<class T>
multivariate_laurentpoly<T> invert_variable(const multivariate_laurentpoly<T>& p, unsigned index = 1) {
multivariate_laurentpoly<T> result;
for (typename map<multivariate_laurent_monomial, T>::const_iter i = p.coeffs; i; i ++) {
multivariate_laurent_monomial mon = i.key();
T c = i.val();
multivariate_laurent_monomial mon2;
for(map<unsigned, int>::const_iter i = mon.m; i; i++) {
if(i.key() == index) {
mon2 *= multivariate_laurent_monomial(VARIABLE, i.key(), -i.val());
}
else {
mon2 *= multivariate_laurent_monomial(VARIABLE, i.key(), i.val());
}
}
result += multivariate_laurentpoly<T>(c, mon2);
}
return result;
}
template<class T>
bool check_przytycki_cong(const multivariate_laurentpoly<T>& pol, const int prime,
const int exponent = 1, const unsigned index = 1) {
multivariate_laurentpoly<T> result;
for(typename map<multivariate_laurent_monomial, T>::const_iter i = pol.coeffs; i; i++) {
multivariate_laurent_monomial mon = i.key();
T c = i.val();
multivariate_laurent_monomial mon2;
for(typename map<unsigned, int>::const_iter i = mon.m; i; i++) {
// still need to handle the coefficient
// i.e. we need to take c % p
if(i.key() == index) {
int v = i.val() % (2 * prime);
if(v < 0) v += (2 * prime);
mon2 *= multivariate_laurent_monomial(VARIABLE, i.key(), v);
c.display_self();
printf("Old: key = %d, val = %d\n", i.key(), i.val());
printf("New: key = %d, val = %d\n", i.key(), v);
}
else
mon2 *= multivariate_laurent_monomial(VARIABLE, i.key(), i.val());
}
result += multivariate_laurentpoly<T>(c, mon2);
}
result.display_self();
return true;
}

88
kk.cpp
View File

@ -251,35 +251,43 @@ compute_gss ()
tex_footer ();
}
multivariate_laurentpoly<Q> compute_jones(const knot_diagram& k, bool reduced) {
cube<Z2> c(kd, reduced);
multivariate_laurentpoly<Q> jones;
for(uint i = 1; i <= c.n_generators; ++i) {
grading gr = c.compute_generator_grading(i);
if(gr.h % 2 == 0) {
jones += multivariate_laurentpoly<Q>(1, VARIABLE, 1, gr.q);
}
else {
jones += multivariate_laurentpoly<Q>(-1, VARIABLE, 1, gr.q);
}
}
return jones;
}
void compute_khp(const knot_diagram& k, bool reduced) {
cube<Z2> c(kd,reduced);
multivariate_laurentpoly<Z> compute_jones(const knot_diagram& k, bool reduced) {
cube<Z2> c(kd, reduced);
ptr<const module<Z2> > C = c.khC;
mod_map<Z2> d = c.compute_d(1, 0, 0, 0, 0);
chain_complex_simplifier<Z2> s(C, d, maybe<int>(1), maybe<int>(0));
C = s.new_C;
d = s.new_d;
C->show_self();
//multivariate_laurentpoly<Q> khp;
//for(uint i = 1; i <= c.n_generators; ++i) {
// grading gr = c.compute_generator_grading(i);
// khp += multivariate_laurentpoly<Q>(1, VARIABLE, gr.h, gr.q);
//}
printf("The Poincare polynomial \n");
multivariate_laurentpoly<Z> jones;
for(uint i = 1; i <= c.n_generators; ++i) {
grading gr = c.compute_generator_grading(i);
if(gr.h % 2 == 0) {
jones += multivariate_laurentpoly<Z>(1, VARIABLE, 1, gr.q);
}
else {
jones += multivariate_laurentpoly<Z>(-1, VARIABLE, 1, gr.q);
}
}
return jones;
}
template<class R>
multivariate_laurentpoly<Z> compute_khp(const knot_diagram& k, bool reduced) {
cube<R> c(kd,reduced);
ptr<const module<R> > C = c.khC;
mod_map<R> d = c.compute_d(1, 0, 0, 0, 0);
chain_complex_simplifier<R> s(C, d, maybe<int>(1), maybe<int>(0));
C = s.new_C;
d = s.new_d;
multivariate_laurentpoly<Z> khp;
for(uint i = 1; i <= C->dim(); ++i) {
grading gr = C->generator_grading(i);
multivariate_laurentpoly<Z> p1 = multivariate_laurentpoly<Z>(1, VARIABLE, 0, gr.h);
multivariate_laurentpoly<Z> p2 = multivariate_laurentpoly<Z>(1, VARIABLE, 1, gr.q);
multivariate_laurentpoly<Z> p3 = p1 * p2;
khp += p3;
}
return khp;
}
template<class R> void
@ -733,16 +741,36 @@ main (int argc, char **argv)
compute_gss ();
}
else if(!strcmp(invariant, "jones")) {
multivariate_laurentpoly<Q> jones_pol = compute_jones(kd, reduced);
printf("Jones polynomial of %s is equal to:\n", knot);
jones_pol.show_self();
printf("\n");
multivariate_laurentpoly<Z> jones_pol = compute_jones(kd, reduced);
printf("Jones polynomial of %s is equal to (coefficients in %s):\n", knot, field);
jones_pol.display_self();
}
else if(!strcmp(invariant, "przytycki_cong")) {
multivariate_laurentpoly<Q> jones_pol = compute_jones(kd, reduced);
// need to add a commandline switch to be able to read
// symmetry order
multivariate_laurentpoly<Z> jones_pol = compute_jones(kd, reduced);
multivariate_laurentpoly<Z> inv_jones_pol = invert_variable(jones_pol,1);
jones_pol.display_self();
inv_jones_pol.display_self();
multivariate_laurentpoly<Z> diff = jones_pol - inv_jones_pol;
diff.display_self();
check_przytycki_cong(diff, 3);
}
else if(!strcmp(invariant, "khp")) {
compute_khp(kd, reduced);
multivariate_laurentpoly<Z> khp;
if(!strcmp(field, "Z2"))
khp = compute_khp<Z2>(kd, reduced);
else if(!strcmp(field, "Z3"))
khp = compute_khp<Zp<3> >(kd, reduced);
else if(!strcmp(field, "Q"))
khp = compute_khp<Q>(kd, reduced);
else
{
fprintf (stderr, "error: unknown field %s\n", field);
exit (EXIT_FAILURE);
}
printf("Khovanov polynomial of %s is equal to (coefficients %s):\n", knot, field);
khp.display_self();
}
else
{