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First shot at periodicity congruences

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Verification of Przytycki's congruence works fine, however there are
some small modifications that could be made.

The, naive, approach to the verification of periodicity congruence for
the Khovanov polynomial didn't work. There are just too many cases to
check. Generation of all of them can exhaust the memory.
This commit is contained in:
Wojciech Politarczyk 2016-11-19 21:01:10 +01:00
parent 66813d4a15
commit 203477c3a8
24 changed files with 800 additions and 288 deletions
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1
.gitignore vendored
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@ -11,3 +11,4 @@ GTAGS
/parallel.cmd.[eo]*
/save
*/save
/org

11
Makefile
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@ -27,7 +27,9 @@ KNOTKIT_OBJS = planar_diagram.o dt_code.o knot_diagram.o cube.o steenrod_square.
knot_parser/knot_parser.o knot_parser/knot_scanner.o \
rd_parser/rd_parser.o rd_parser/rd_scanner.o
COMMON_OBJS = $(KNOTKIT_OBJS) $(ALGEBRA_OBJS) $(LIB_OBJS)
PERIODICITY_OBJS = periodicity.o
COMMON_OBJS = $(KNOTKIT_OBJS) $(ALGEBRA_OBJS) $(LIB_OBJS) $(PERIODICITY_OBJS)
LIB_HEADERS = lib/lib.h lib/show.h lib/refcount.h lib/pair.h lib/maybe.h lib/vector.h \
lib/set_wrapper.h lib/set.h lib/hashset.h \
@ -44,6 +46,8 @@ KNOTKIT_HEADERS = knotkit.h planar_diagram.h dt_code.h knot_diagram.h \
smoothing.h cobordism.h cube.h steenrod_square.h \
spanning_tree_complex.h cube_impl.h sseq.h simplify_chain_complex.h
PERIODICITY_HEADERS = periodicity.h
LIBS = -lgmpxx -lgmp -lz
all: kk
@ -55,7 +59,7 @@ all: kk
$(CXX) -c $(CXXFLAGS) $< -o $@
kk: kk.o $(COMMON_OBJS)
$(CXX) $(LDFLAGS) -o kk $^ $(LIBS)
$(CXX) $(LDFLAGS) -pg -o kk $^ $(LIBS)
main: main.o $(COMMON_OBJS)
$(CXX) $(LDFLAGS) -o main $^ $(LIBS)
@ -115,6 +119,7 @@ realclean: clean
$(LIB_OBJS): $(LIB_HEADERS)
$(ALGEBRA_OBJS): $(ALGEBRA_HEADERS) $(LIB_HEADERS)
$(KNOTKIT_OBJS) main.o mpimain.o kk.o: $(KNOTKIT_HEADERS) $(ALGEBRA_HEADERS) $(LIB_HEADERS)
$(KNOTKIT_OBJS) main.o mpimain.o kk.o: $(KNOTKIT_HEADERS) $(ALGEBRA_HEADERS) $(LIB_HEADERS) $(PERIODICITY_HEADERS)
$(PERIODICITY_OBJS) : $(PERIODICITY_HEADERS)
mpimain.o mpi_aux.o: mpi_aux.h

48
algebra/Q.h
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@ -13,47 +13,8 @@ class Q
using linear_combination_const_iter = ::linear_combination_const_iter<Q>;
private:
// enum steal { STEAL };
// enum copy { COPY };
// class Q_impl : public refcounted
// {
// public:
// mpq_t x;
// public:
// Q_impl () { mpq_init (x); }
// Q_impl (int init)
// {
// mpq_init (x);
// mpq_set_si (x, init, 1);
// }
// Q_impl (copy, mpq_srcptr init)
// {
// mpq_init (x);
// mpq_set (x, init);
// }
// Q_impl (steal, mpq_srcptr init) { x[0] = *init; }
// Q_impl (reader &r)
// {
// mpq_init (x);
// r.read_mpz (mpq_numref (x));
// r.read_mpz (mpq_denref (x));
// }
// ~Q_impl () { mpq_clear (x); }
// void write_self (writer &w) const
// {
// w.write_mpz (mpq_numref (x));
// w.write_mpz (mpq_denref (x));
// }
// };
// ptr<Q_impl> impl;
std::shared_ptr<mpq_class> impl;
void write_state() const {
void write_state() const {
std::cout << "I store the following value " << *this << "\n";
std::cout << "Number of objects pointing to the same value " << impl.use_count() << "\n";
/* std::cout << "I point to " << impl.get() << "\n"; */
@ -246,6 +207,11 @@ class Q
int get_count() const {
return impl.use_count();
}
Z get_num() const {
return Z(impl.get()->get_num());
}
Z get_den() const {
return Z(impl.get()->get_den());
}
};
#endif // _KNOTKIT_ALGEBRA_Q_H

23
algebra/Z.h
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@ -145,6 +145,24 @@ class Z
}
}
Z operator % (const Z& z) const {
if(Z(0) < z) {
// return Z(*impl.get() % *z.impl.get());
mpz_class r;
mpz_fdiv_r(r.get_mpz_t(), impl.get()->get_mpz_t(), z.impl.get()->get_mpz_t());
return Z(r);
}
else
return *this;
}
Z operator % (unsigned p) const {
if(p != 0)
return operator % (Z(p));
else
return *this;
}
Z recip () const {
assert(is_unit());
return *this;
@ -234,6 +252,9 @@ class Z
int get_count() const {
return impl.use_count();
}
};
unsigned get_ui() const {
return impl.get()->get_ui();
}
};
#endif // _KNOTKIT_ALGEBRA_Z_H

4
algebra/Z2.h
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@ -1,5 +1,6 @@
#ifndef KNOTKIT_ALGEBRA_Z2_H
#define KNOTKIT_ALGEBRA_Z2_H
#include <algebra/Z.h>
#include <iostream>
#include <tuple>
#include <cassert>
@ -8,7 +9,6 @@ class Z2
{
public:
using linear_combination = ::linear_combination<Z2>;
// typedef linear_combination_iter<Z2> linear_combination_iter;
using linear_combination_const_iter = ::linear_combination_const_iter<Z2>;
private:
@ -24,6 +24,7 @@ class Z2
x.v = 0;
}
Z2 (reader &r) { v = r.read_bool (); }
Z2 (const Z& z) : v((z % 2).get_ui()) {}
~Z2 () { }
Z2& operator = (const Z2& x) { v = x.v; return *this; }
@ -88,5 +89,4 @@ class Z2
void show_self () const { std::cout << *this; }
void display_self () const { std::cout << *this << "\n"; }
};
#endif //KNOTKIT_ALGEBRA_Z2_H

4
algebra/Zp.h
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@ -1,6 +1,6 @@
#ifndef KNOTKIT_ALGEBRA_ZP_H
#define KNOTKIT_ALGEBRA_ZP_H
#include <algebra/Z.h>
#include <iostream>
#include <tuple>
#include <cassert>
@ -32,6 +32,7 @@ class Zp
Zp (Zp&& x) : v(std::move(x.v)) {
x.v = 0;
}
Zp (const Z& z) : v((z % p).get_ui()) {}
~Zp () { }
Zp& operator = (const Zp& x) { v = x.v; return *this; }
@ -123,5 +124,4 @@ class Zp
void show_self () const { std::cout << v << "(" << p << ")"; }
void display_self () const { std::cout << *this << "\n"; }
};
#endif //KNOTKIT_ALGEBRA_ZP_H

4
algebra/algebra.h
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@ -1,3 +1,5 @@
#ifndef _KNOTKIT_ALGEBRA_H
#define _KNOTKIT_ALGEBRA_H
#include <lib/lib.h>
#ifdef DEBUG_ALGEBRA
@ -73,3 +75,5 @@ enum variable { VARIABLE };
#include <algebra/module.h>
#include <algebra/linear_combination.h>
#endif // _KNOTKIT_ALGEBRA_H

4
algebra/fraction_field.h
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@ -1,3 +1,5 @@
#ifndef _KNOTKIT_ALGEBRA_FRACTION_FIELD_H
#define _KNOTKIT_ALGEBRA_FRACTION_FIELD_H
template<class T> class fraction_field
{
@ -243,3 +245,5 @@ fraction_field<T>::show_self () const
printf (")");
}
}
#endif // _KNOTKIT_ALGEBRA_FRACTION_FIELD_H

14
algebra/grading.h
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@ -1,4 +1,5 @@
#ifndef _KNOTKIT_ALGEBRA_GRADINGS_H
#define _KNOTKIT_ALGEBRA_GRADINGS_H
class grading
{
public:
@ -8,6 +9,9 @@ class grading
grading () : h(0), q(0) { }
grading (int h_, int q_) : h(h_), q(q_) { }
grading (const grading &gr) : h(gr.h), q(gr.q) { }
grading (grading&& gr) : h(std::move(gr.h)), q(std::move(gr.q)) {
gr.h = gr.q = 0;
}
grading (reader &r)
{
h = r.read_int ();
@ -17,6 +21,12 @@ class grading
~grading () { }
grading &operator = (const grading &gr) { h = gr.h; q = gr.q; return *this; }
grading& operator = (grading&& gr) {
h = std::move(gr.h);
q = std::move(gr.q);
gr.h = gr.q = 0;
return *this;
}
grading operator + (const grading &gr) const
{
@ -61,3 +71,5 @@ class grading
void show_self () const;
void display_self () const;
};
#endif // _KNOTKIT_ALGEBRA_GRADINGS_H

5
algebra/linear_combination.h
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@ -1,4 +1,5 @@
#ifndef _KNOTKIT_ALGEBRA_LINEAR_COMBINATIONS_H
#define _KNOTKIT_ALGEBRA_LINEAR_COMBINATIONS_H
template<class R> class linear_combination_const_iter;
template<class R>
@ -512,3 +513,5 @@ class linear_combination_const_iter<Z2>
unsigned key () const { return i.val (); }
Z2 val () { return Z2 (1); }
};
#endif // _KNOTKIT_ALGEBRA_LINEAR_COMBINATIONS_H

23
algebra/module.h
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@ -1,4 +1,5 @@
#ifndef _KNOTKIT_ALGEBRA_MODULE_H
#define _KNOTKIT_ALGEBRA_MODULE_H
template<class R> class mod_map;
template<class R> class mod_span;
@ -85,8 +86,10 @@ class module : public refcounted
ptr<const quotient_module<R> > quotient (ptr<const free_submodule<R> > m) const;
ptr<const free_submodule<R> > submodule (const mod_span<R> &span) const;
multivariate_laurentpoly<Z> free_poincare_polynomial () const;
template<unsigned p>
multivariate_laurentpoly<Zp<p>> free_poincare_polynomial () const;
multivariate_laurentpoly<Z> free_delta_poincare_polynomial () const;
multivariate_laurentpoly<Z> free_ell_poincare_polynomial () const;
@ -1476,6 +1479,20 @@ module<R>::free_poincare_polynomial () const
return r;
}
template<class R> template <unsigned p> multivariate_laurentpoly<Zp<p>>
module<R>::free_poincare_polynomial() const {
multivariate_laurentpoly<Zp<p>> r;
for (unsigned i = 1; i <= free_rank (); i ++)
{
grading hq = generator_grading (i);
multivariate_laurent_monomial m;
m.push_exponent (1, hq.h);
m.push_exponent (2, hq.q);
r.muladdeq (1, m);
}
return r;
}
template<class R> multivariate_laurentpoly<Z>
module<R>::free_delta_poincare_polynomial () const
{
@ -2130,3 +2147,5 @@ reader::read_mod ()
return module<R>::reader_id_module(id);
}
}
#endif // _KNOTKIT_ALGEBRA_MODULE_H

182
algebra/multivariate_laurentpoly.h
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@ -1,4 +1,7 @@
#ifndef _KNOTKIT_ALGEBRA_MULTIVARIATE_LAURENPOLY_H
#define _KNOTKIT_ALGEBRA_MULTIVARIATE_LAURENPOLY_H
#include <sstream>
#include <string>
/* multivariate polynomial in a (vector) variable x with coefficients
in T. */
@ -133,17 +136,31 @@ class multivariate_laurent_monomial
if (e != 0)
m.push (j, e);
}
std::string to_string() const {
std::ostringstream res;
for (map<unsigned, int>::const_iter i = m; i; i ++) {
assert (i.val () != 0);
if (i.val () == 1) {
res << "x" << std::to_string(i.key());
}
else {
res << "x"
<< std::to_string(i.key())
<< "^"
<< std::to_string(i.val());
}
}
return res.str();
}
friend std::ostream& operator << (std::ostream& os, const multivariate_laurent_monomial& m) {
return os << m.to_string();
}
void show_self () const
{
for (map<unsigned, int>::const_iter i = m; i; i ++)
{
assert (i.val () != 0);
if (i.val () == 1)
printf ("x%d", i.key ());
else
printf ("x%d^%d", i.key (), i.val ());
}
std::cout << *this;
}
void write_self (writer &w) const { write (w, m); }
@ -161,12 +178,11 @@ template<class T>
class multivariate_laurentpoly
{
public:
typedef ::linear_combination<multivariate_laurentpoly<T> > linear_combination;
typedef ::linear_combination_const_iter<multivariate_laurentpoly<T> >
linear_combination_const_iter;
using linear_combination = ::linear_combination<multivariate_laurentpoly<T>>;
using linear_combination_const_iter = ::linear_combination_const_iter<multivariate_laurentpoly<T> >;
public:
typedef multivariate_laurent_monomial monomial;
using monomial = multivariate_laurent_monomial;
map<monomial, T> coeffs;
@ -216,13 +232,15 @@ class multivariate_laurentpoly
if (c != 0)
coeffs.push (m, c);
}
template<class U>
multivariate_laurentpoly (const multivariate_laurentpoly<U>& pol);
multivariate_laurentpoly (const multivariate_laurentpoly &p) : coeffs(p.coeffs) { }
multivariate_laurentpoly (copy, const multivariate_laurentpoly &p)
// ??? COPY2?
: coeffs(COPY, p.coeffs)
{
}
{}
multivariate_laurentpoly (reader &r) : coeffs(r) { }
@ -294,6 +312,7 @@ class multivariate_laurentpoly
unsigned card () const { return coeffs.card (); }
pair<monomial, T> head () const { return coeffs.head (); }
pair<monomial, T> tail() const { return coeffs.tail(); }
multivariate_laurentpoly &operator += (const multivariate_laurentpoly &p);
multivariate_laurentpoly &operator -= (const multivariate_laurentpoly &p);
@ -345,18 +364,33 @@ class multivariate_laurentpoly
#ifndef NDEBUG
void check () const;
#endif
multivariate_laurentpoly evaluate(T val, unsigned index) const;
static void show_ring ()
{
T::show_ring ();
printf ("[x_1^+/-1, ..., x_n^+/-1]");
}
void display_self () const { show_self (); newline (); }
void show_self () const;
std::string to_string() const;
void display_self () const {
std::cout << *this << "\n";
}
void show_self () const {
std::cout << *this;
}
void write_self (writer &w) const { write (w, coeffs); }
};
template<class T> template<class U>
multivariate_laurentpoly<T>::multivariate_laurentpoly(const multivariate_laurentpoly<U>& pol) {
for(typename map<monomial, U>::const_iter i = pol.coeffs; i; i++) {
if(T(i.val()) != 0)
coeffs.push(i.key(), T(i.val()));
}
}
template<class T> multivariate_laurentpoly<T>
operator * (const T &s, const multivariate_laurentpoly<T> &p)
{
@ -444,42 +478,50 @@ multivariate_laurentpoly<T>::check () const
}
#endif
template<class T> void
multivariate_laurentpoly<T>::show_self () const
{
template<class T>
std::string multivariate_laurentpoly<T>::to_string() const {
std::ostringstream res;
unsigned first = 1;
for (typename map<monomial, T>::const_iter i = coeffs; i; i ++)
{
monomial m = i.key ();
T c = i.val ();
for (typename map<monomial, T>::const_iter i = coeffs; i; i ++) {
monomial m = i.key ();
T c = i.val ();
assert (c != 0);
if (first)
first = 0;
else
res << " + ";
assert (c != 0);
if (first)
first = 0;
if (m == 1) {
if (c == 1)
res << "1";
else
printf (" + ");
if (m == 1)
{
if (c == 1)
printf ("1");
else
show (c);
}
else
{
if (c != 1)
{
show (c);
printf ("*");
}
show (m);
}
res << c;
}
else {
if (c != 1) {
res << c << "*";
}
res << m;
}
}
if (first)
printf ("0");
res << "0";
return res.str();
}
template<class T>
std::ostream& operator << (std::ostream& os, const multivariate_laurentpoly<T>& pol) {
return os << pol.to_string();
}
template<class T, class U>
multivariate_laurentpoly<T> reduce(const multivariate_laurentpoly<U>& pol) {
multivariate_laurentpoly<T> res;
for(typename map<unsigned, U>::const_iter i = pol.coeffs; i; i++) {
res += multivariate_laurentpoly<T>(i.val(), i.key());
}
return res;
}
// functions below were added to verify several periodicity criteria
@ -507,29 +549,23 @@ multivariate_laurentpoly<T> invert_variable(const multivariate_laurentpoly<T>& p
}
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());
multivariate_laurentpoly<T>
multivariate_laurentpoly<T>::evaluate(T val, unsigned index) const {
using polynomial = multivariate_laurentpoly<T>;
using monomial = multivariate_laurent_monomial;
polynomial res;
for(typename map<monomial, T>::const_iter i = coeffs; i; i++) {
if(i.key().m % index) {
int exp = i.key().m[index];
monomial mon = i.key();
mon.m.yank(index);
polynomial temp = polynomial(i.val() * pow(val, exp), mon);
res += temp;
}
result += multivariate_laurentpoly<T>(c, mon2);
else
res += polynomial(i.val(), i.key());
}
result.display_self();
return true;
return res;
}
#endif // _KNOTKIT_ALGEBRA_MULTIVARIATE_LAURENPOLY_H

5
algebra/multivariate_polynomial.h
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@ -1,4 +1,5 @@
#ifndef _KNOTKIT_ALGEBRA_MULTIVARIATE_POLYNOMIAL_H
#define _KNOTKIT_ALGEBRA_MULTIVARIATE_POLYNOMIAL_H
/* multivariate polynomial in a (vector) variable x with coefficients
in T. */
@ -499,3 +500,5 @@ multivariate_polynomial<T, n>::show_self () const
if (first)
printf ("0");
}
#endif // _KNOTKIT_ALGEBRA_MULTIVARIATE_POLYNOMIAL_H

5
algebra/polynomial.h
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@ -1,4 +1,5 @@
#ifndef _KNOTKIT_ALGEBRA_POLYNOMIAL_H
#define _KNOTKIT_ALGEBRA_POLYNOMIAL_H
/* univariate polynomial in a single variable `x' with coefficients in
T. */
@ -512,3 +513,5 @@ class polynomial<Z2>
void display_self () const { show_self (); newline (); }
void show_self () const;
};
#endif // _KNOTKIT_ALGEBRA_POLYNOMIAL_H

434
kk.cpp
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@ -1,4 +1,8 @@
#include <knotkit.h>
#include <periodicity.h>
#include <fstream>
#include <vector>
#include <utility>
const char *program_name;
@ -70,9 +74,10 @@ void tex_footer ()
const char *knot = 0;
const char *invariant = 0;
const char *field = "Z2";
std::string periodicity_test = "Przytycki";
int period = 5;
knot_diagram kd;
bool reduced = 0;
class hg_grading_mapper
@ -250,45 +255,197 @@ compute_gss ()
tex_footer ();
}
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));
template<class R>
multivariate_laurentpoly<Z> compute_khp(const knot_diagram& k, bool reduced = false) {
cube<R> c (kd, reduced);
ptr<const module<R> > C = c.khC;
mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
unsigned m = kd.num_components ();
hg_grading_mapper mapper (m);
chain_complex_simplifier<R> s (C, d,
maybe<int> (1), maybe<int> (0));
C = s.new_C;
d = s.new_d;
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;
return C->free_poincare_polynomial();
}
multivariate_laurentpoly<Z> compute_jones(const knot_diagram& k, bool reduced = false) {
return compute_khp<Z2>(k, reduced).evaluate(-1,1);
}
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;
int compute_s_inv(knot_diagram& kd) {
unsigned m = kd.num_components ();
if (m != 1) {
fprintf (stderr, "error: s-invariant only defined for knots\n");
exit (EXIT_FAILURE);
}
return khp;
cube<R> c (kd, 0);
ptr<const module<R> > C = c.khC;
mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
for (unsigned i = 1; i <= kd.n_crossings; i ++)
d = d + c.H_i (i);
assert (d.compose (d) == 0);
int k = 0;
for (;;) {
chain_complex_simplifier<R> s (C, d,
maybe<int> (1),
maybe<int> (2*k));
C = s.new_C;
d = s.new_d;
k ++;
if (d == 0)
break;
}
assert (C->dim () == 2);
grading gr1 = C->generator_grading (1),
gr2 = C->generator_grading (2);
C->free_poincare_polynomial().display_self();
int qmin = gr1.q,
qmax = gr2.q;
if (qmax < qmin)
std::swap (qmin, qmax);
assert (qmax == qmin + 2);
return qmin + 1;
}
void check_periodicity(std::string out_file) {
if(period == 2 || period == 3) {
std::cout << "Sorry, the criteria don't work for period "
<< period << "...\n";
exit(EXIT_FAILURE);
}
auto result = std::find(primes_list.begin(), primes_list.end(), period);
if(result == primes_list.end()) {
std::cout << "For now you can only check periodicity for primes up to 31..." << "\n";
exit(EXIT_FAILURE);
}
std::ofstream out(out_file);
if(periodicity_test == "all") {
std::cout << "I will perform both test..." << "\n";
}
else if(periodicity_test == "Przytycki") {
switch(period) {
case 5: {
Przytycki_periodicity_checker<5> pcc;
if(out_file.size() != 0)
out << pcc(compute_jones(kd)) << std::endl;
else
std::cout << pcc(compute_jones(kd)) << std::endl;
break;
}
case 7: {
Przytycki_periodicity_checker<7> pcc;
if(out_file.size() != 0)
out << pcc(compute_jones(kd)) << std::endl;
else
std::cout << pcc(compute_jones(kd)) << std::endl;
break;
}
case 11: {
Przytycki_periodicity_checker<11> pcc;
if(out_file.size() != 0)
out << pcc(compute_jones(kd)) << std::endl;
else
std::cout << pcc(compute_jones(kd)) << std::endl;
break;
}
case 13: {
Przytycki_periodicity_checker<13> pcc;
if(out_file.size() != 0)
out << pcc(compute_jones(kd)) << std::endl;
else
std::cout << pcc(compute_jones(kd)) << std::endl;
break;
}
case 17: {
Przytycki_periodicity_checker<17> pcc;
if(out_file.size() != 0)
out << pcc(compute_jones(kd)) << std::endl;
else
std::cout << pcc(compute_jones(kd)) << std::endl;
break;
}
case 19: {
Przytycki_periodicity_checker<19> pcc;
if(out_file.size() != 0)
out << pcc(compute_jones(kd)) << std::endl;
else
std::cout << pcc(compute_jones(kd)) << std::endl;
break;
}
}
}
else if(periodicity_test == "Kh") {
Kh_periodicity_checker pc(kd);
if(out_file.size() != 0)
out << pc(period) << std::endl;
else
std::cout << pc(period) << std::endl;
}
else {
std::cout << "Sorry, I don't recognize this option..." << "\n";
exit(EXIT_FAILURE);
}
}
// template<class R>
// void check_periodicity_criterion(knot_diagram kd, int prime = 5, Test_type t){
// using monomial = multivariate_laurent_monomial;
// using polynomial = multivariate_laurentpoly<Z>;
// unsigned m = kd.num_components();
// if(m != 1) {
// std::cerr << "Error: for now this only works for knots\n";
// exit(EXIT_FAILURE);
// }
// polynomial khp, lee_p;
// // Adding braces so that the cube can be destroyed
// // when its not needed anymore
// {
// cube<R> c(kd,0);
// ptr<const module<R>> C = c.khC;
// mod_map<R> d = c.compute_d(1, 0, 0, 0, 0);
// for(unsigned i = 1; i <= kd.n_crossings; i++)
// d = d + c.H_i(i);
// assert (d.compose(d) == 0);
// // computing Khovanov homology
// if(verbose)
// std::cout << "Computing Khovanov polynomial..." << "\n";
// chain_complex_simplifier<R> s(C, d, maybe<int>(1), maybe<int>(0));
// C = s.new_C;
// d = s.new_d;
// khp = C->free_poincare_polynomial();
// // computing Lee homology
// if(verbose)
// std::cout << "Computing Lee polynomial..." << "\n";
// chain_complex_simplifier<R> s1(C, d, maybe<int>(1), maybe<int>(2));
// C = s1.new_C;
// d = s1.new_d;
// assert(C->dim() == 2);
// lee_p = C->free_poincare_polynomial();
// }
// if(verbose) {
// std::cout << "Khovanov polynomial = "
// << khp << "\n"
// << "Lee polynomial = "
// << lee_p << "\n";
// }
// periodicity_checker pc(khp, lee_p, prime, t);
// pc();
// }
template<class R> void
compute_invariant ()
{
@ -413,48 +570,10 @@ compute_invariant ()
tex_footer ();
}
else if (!strcmp (invariant, "s"))
{
unsigned m = kd.num_components ();
if (m != 1)
{
fprintf (stderr, "error: s-invariant only defined for knots\n");
exit (EXIT_FAILURE);
}
cube<R> c (kd, 0);
ptr<const module<R> > C = c.khC;
mod_map<R> d = c.compute_d (1, 0, 0, 0, 0);
for (unsigned i = 1; i <= kd.n_crossings; i ++)
d = d + c.H_i (i);
assert (d.compose (d) == 0);
int k = 0;
for (;;)
{
chain_complex_simplifier<R> s (C, d,
maybe<int> (1), maybe<int> (2*k));
C = s.new_C;
d = s.new_d;
k ++;
if (d == 0)
break;
}
assert (C->dim () == 2);
grading gr1 = C->generator_grading (1),
gr2 = C->generator_grading (2);
int qmin = gr1.q,
qmax = gr2.q;
if (qmax < qmin)
std::swap (qmin, qmax);
assert (qmax == qmin + 2);
fprintf (outfp, "s(%s; %s) = %d\n", knot, field, qmin + 1);
}
{
int s = compute_s_inv<R>(kd);
fprintf (outfp, "s(%s; %s) = %d\n", knot, field, s);
}
else
{
fprintf (stderr, "error: unknown invariant %s\n", invariant);
@ -617,69 +736,73 @@ main (int argc, char **argv)
const char *file = 0;
for (int i = 1; i < argc; i ++)
{
if (argv[i][0] == '-')
{
if (strcmp (argv[i], "-r") == 0)
reduced = 1;
else if (strcmp (argv[i], "-h") == 0)
{
usage ();
exit (EXIT_SUCCESS);
}
else if (strcmp (argv[i], "-demo") == 0)
{
run_demo();
exit (EXIT_SUCCESS);
}
else if (!strcmp (argv[i], "-v"))
verbose = 1;
else if (!strcmp (argv[i], "-f"))
{
i ++;
if (i == argc)
{
fprintf (stderr, "error: missing argument to option `-f'\n");
exit (EXIT_FAILURE);
}
field = argv[i];
}
else if (!strcmp (argv[i], "-o"))
{
i ++;
if (i == argc)
{
fprintf (stderr, "error: missing argument to option `-o'\n");
exit (EXIT_FAILURE);
}
file = argv[i];
}
else
{
fprintf (stderr, "error: unknown argument `%s'\n", argv[1]);
fprintf (stderr, " use -h for usage\n");
exit (EXIT_FAILURE);
}
for (int i = 1; i < argc; i ++) {
if (argv[i][0] == '-') {
if (strcmp (argv[i], "-r") == 0)
reduced = 1;
else if (strcmp (argv[i], "-h") == 0) {
usage ();
exit (EXIT_SUCCESS);
}
else if (strcmp (argv[i], "-demo") == 0) {
run_demo();
exit (EXIT_SUCCESS);
}
else if (!strcmp (argv[i], "-v"))
verbose = 1;
else if (!strcmp (argv[i], "-f")) {
i ++;
if (i == argc) {
fprintf (stderr, "error: missing argument to option `-f'\n");
exit (EXIT_FAILURE);
}
else
{
if (knot)
{
fprintf (stderr, "error: too many arguments\n");
fprintf (stderr, " use -h for usage\n");
exit (EXIT_FAILURE);
}
else if (invariant)
knot = argv[i];
else
{
assert (invariant == 0);
invariant = argv[i];
}
field = argv[i];
}
else if (!strcmp (argv[i], "-o")) {
i ++;
if (i == argc) {
fprintf (stderr, "error: missing argument to option `-o'\n");
exit (EXIT_FAILURE);
}
file = argv[i];
}
else if(!strcmp (argv[i], "-p")) {
i++;
if(i == argc) {
fprintf (stderr, "error: missing argument to option `-o'\n");
exit (EXIT_FAILURE);
}
period = std::stoi(argv[i]);
}
else if (!strcmp(argv[i], "-t")) {
i++;
if(i == argc) {
fprintf (stderr, "error: missing argument to option `-o'\n");
exit (EXIT_FAILURE);
}
periodicity_test = argv[i];
}
else {
fprintf (stderr, "error: unknown argument `%s'\n", argv[1]);
fprintf (stderr, " use -h for usage\n");
exit (EXIT_FAILURE);
}
}
else {
if (knot) {
fprintf (stderr, "error: too many arguments\n");
fprintf (stderr, " use -h for usage\n");
exit (EXIT_FAILURE);
}
else if (invariant)
knot = argv[i];
else {
assert (invariant == 0);
invariant = argv[i];
}
}
}
if (!knot)
{
fprintf (stderr, "error: too few arguments, <invariant> or <knot> missing\n");
@ -687,15 +810,13 @@ main (int argc, char **argv)
exit (EXIT_FAILURE);
}
if (file)
{
outfp = fopen (file, "w");
if (!outfp)
{
stderror ("fopen: %s", file);
exit (EXIT_FAILURE);
}
if (file) {
outfp = fopen (file, "w");
if (!outfp) {
stderror ("fopen: %s", file);
exit (EXIT_FAILURE);
}
}
else
outfp = stdout;
@ -740,36 +861,31 @@ main (int argc, char **argv)
compute_gss ();
}
else if(!strcmp(invariant, "jones")) {
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();
std::cout << "Jones polynomial of " << knot << " = " << compute_jones(kd, reduced) << "\n";
}
else if(!strcmp(invariant, "przytycki_cong")) {
// 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, "periodicity")) {
check_periodicity((file ? std::string(file) : std::string()));
}
else if(!strcmp(invariant, "khp")) {
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);
khp = compute_khp<Zp<3>>(kd, reduced);
else if(!strcmp(field, "Z5"))
khp = compute_khp<Zp<5>>(kd, reduced);
else if (!strcmp(field, "Z7"))
khp = compute_khp<Zp<7>>(kd,reduced);
else if(!strcmp(field, "Q"))
khp = compute_khp<Q>(kd, reduced);
else
{
fprintf (stderr, "error: unknown field %s\n", field);
std::cerr << "Unknown field: " << field << std::endl;
exit (EXIT_FAILURE);
}
printf("Khovanov polynomial of %s is equal to (coefficients %s):\n", knot, field);
khp.display_self();
std::cout << "Khovanov polynomial (coefficients in " << field
<< ") of " << knot << " = " << std::endl
<< khp << std::endl;
}
else
{

6
knot_diagram.h
View File

@ -1,3 +1,7 @@
#ifndef _KNOTKIT_KNOT_DIAGRAM_H
#define _KNOTKIT_KNOT_DIAGRAM_H
#include <planar_diagram.h>
// for building knot_diagram
inline unsigned edge_from_ept (unsigned e)
@ -183,3 +187,5 @@ class knot_diagram
void show_self () const;
void display_self () const;
};
#endif // _KNOTKIT_KNOT_DIAGRAM_H

4
knotkit.h
View File

@ -1,3 +1,5 @@
#ifndef _KNOTKIT_KNOTKIT_H
#define _KNOTKIT_KNOTKIT_H
// includes lib.h
#include <algebra/algebra.h>
@ -123,3 +125,5 @@ resolution_diagram parse_resolution_diagram (const char *s);
// 11 <= n <= 15
basedvector<basedvector<unsigned, 1>, 1> mutant_knot_groups (unsigned n);
#endif // _KNOTKIT_KNOTKIT_H

7
lib/bitset.h
View File

@ -1,3 +1,8 @@
#ifndef _KNOTKIT_LIB_BITSET_H
#define _KNOTKIT_LIB_BITSET_H
#include <lib/lib.h>
#include <lib/vector.h>
typedef uint64 word_t;
@ -136,3 +141,5 @@ bitset::operator % (unsigned i) const
unsigned b = (i - 1) & word_bit_mask;
return word_bittest (v[w], b + 1);
}
#endif // _KNOTKIT_LIB_BITSET_H

4
lib/lib.h
View File

@ -1,3 +1,5 @@
#ifndef _KNOTKIT_LIB_LIB_H
#define _KNOTKIT_LIB_LIB_H
#include <assert.h>
#include <stdlib.h>
@ -235,3 +237,5 @@ public:
map<unsigned, unsigned> io_id_id;
};
#endif // _KNOTKIT_LIB_LIB_H

6
lib/vector.h
View File

@ -1,3 +1,7 @@
#ifndef _KNOTKIT_LIB_VECTOR_H
#define _KNOTKIT_LIB_VECTOR_H
#include <lib/lib.h>
template<class T>
class vector
@ -386,3 +390,5 @@ public:
unsigned lower_bound (const T &v) const { return vector<T>::lower_bound (v) + B; }
unsigned upper_bound (const T &v) const { return vector<T>::upper_bound (v) + B; }
};
#endif // _KNOTKIT_LIB_VECTOR_H

158
periodicity.cpp Normal file
View File

@ -0,0 +1,158 @@
#include <periodicity.h>
#include <simplify_chain_complex.h>
void Kh_periodicity_checker::compute_knot_polynomials(knot_diagram& kd) {
unsigned m = kd.num_components ();
if (m != 1) {
std::cerr << "warning: this implementation of the criterion works for knots only...";
exit (EXIT_FAILURE);
}
cube<Z2> c (kd, 0);
ptr<const module<Z2> > C = c.khC;
mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
for (unsigned i = 1; i <= kd.n_crossings; i ++)
d = d + c.H_i (i);
assert (d.compose (d) == 0);
// computing Khovanov homology
if(verbose)
std::cout << "Computing Khovanov homology" << std::endl;
{
chain_complex_simplifier<Z2> s (C, d, maybe<int>(1), maybe<int>(0));
C = s.new_C;
d = s.new_d;
khp = C->free_poincare_polynomial();
if(verbose)
std::cout << "KhP = " << khp << "\n";
}
// computing Lee homolgy
if(verbose)
std::cout << "Computing Lee homology" << std::endl;
{
chain_complex_simplifier<Z2> s(C, d, maybe<int>(1), maybe<int>(2));
C = s.new_C;
d = s.new_d;
leep = C->free_poincare_polynomial();
if(d != 0) {
std::cout << "For now, you can only use this criterion on Kh-thin knots." << std::endl;
exit(EXIT_FAILURE);
}
if(verbose) {
std::cout << "LeeP = " << leep << "\n";
}
}
}
void Kh_periodicity_checker::compute_quot() {
polynomial diff = khp - leep;
while(diff != 0) {
pair<monomial, Z> m = diff.head();
if(m.first.m[1] == 1) {
pair<monomial, Z> m1 = diff.tail();
while(m1.first.m.card() == 1 && m1.first.m[2]) {
quot += polynomial(m1.second, m1.first);
polynomial p = polynomial(m1.second, m1.first) * mul;
diff -= p;
m1 = diff.tail();
}
}
quot += polynomial(m.second, m.first);
polynomial p = polynomial(m.second, m.first) * mul;
diff -= p;
}
}
std::pair<multivariate_laurentpoly<Z>, multivariate_laurentpoly<Z>>
Kh_periodicity_checker::compute_quotient_and_remainder(const polynomial& quot,
int period) const {
polynomial quotient, remainder;
for(map<monomial, Z>::const_iter i = quot.coeffs; i; i++) {
std::tuple<Z,Z> div = i.val().divide_with_remainder(period - 1);
quotient += polynomial(std::get<0>(div), i.key());
remainder += polynomial(std::get<1>(div), i.key());
}
if(verbose) {
std::cout << "Decomposition of Khp = " << std::endl
<< leep << " + ("
<< mul << ") * ("
<< remainder;
if(quotient != 0) {
std::cout << " + " << (period - 1)
<< " * (" << quotient
<< ")";
}
std::cout << ")" << std::endl;
}
return std::make_pair(quotient, remainder);
}
std::list<multivariate_laurentpoly<Z>>
Kh_periodicity_checker::generate_candidates(const polynomial &q) const {
std::list<polynomial> result;
Z size = 0;
map<monomial, Z>::const_iter i = q.coeffs;
for(int j = 0; Z(j) <= i.val(); j++) {
result.push_back(polynomial(Z(j), i.key()));
size += 1;
}
i++;
for( ; i; i++) {
for(int j = 1; Z(j) <= i.val(); j++) {
std::list<polynomial> temp_list;
for_each(result.begin(), result.end(),
[&result, &temp_list, j, i](const polynomial& p){
temp_list.push_back(p + polynomial(Z(j), i.key()));
});
result.splice(result.end(), temp_list);
size += 1;
}
}
return result;
}
bool Kh_periodicity_checker::check(const polynomial& q,
const polynomial& r,
int period) const {
periodic_congruence_checker<Z> pcc(period);
polynomial t = leep + mul * r;
if(q == 0) {
return pcc(t.evaluate(-1,1));
}
else {
// generate all polynomials
if(verbose) {
std::cout << "Generating all candidates..." << std::endl;
}
std::list<polynomial> candidates = generate_candidates(q);
// Z i = 0;
// for(auto& l : candidates) {
// i += 1;
// std::cout << i << ": " << l << std::endl;
// }
// and check each one of them
if(verbose)
std::cout << "Checking congruences..." << std::endl;
polynomial m = mul;
return any_of(candidates.begin(), candidates.end(),
[&pcc, &t, &m, &period](const polynomial& p){
return pcc((t + Z(period - 1) * m * p).evaluate(-1,1));
});
}
}
std::string Kh_periodicity_checker::operator () (int period) const {
std::ostringstream out;
std::pair<polynomial, polynomial> q_r = compute_quotient_and_remainder(quot, period);
bool res = check(std::get<0>(q_r), std::get<1>(q_r), period);
out << knot << ": period = " << period << ": "
<< (res ? "Maybe" : "No");
return out.str();
}

124
periodicity.h Normal file
View File

@ -0,0 +1,124 @@
#ifndef _KNOTKIT_PERIODICITY_H
#define _KNOTKIT_PERIODICITY_H
#include <knotkit.h>
#include <sstream>
#include <string>
#include <vector>
#include <utility>
#include <list>
#include <algorithm>
extern bool verbose;
extern const char* knot;
extern std::string periodicity_test;
const std::vector<int> primes_list = {5, 7, 11, 13, 17, 19};
enum class Test_type {
Przytycki_criterion,
BKP_criterion,
all
};
template<class T>
class periodic_congruence_checker {
public:
using polynomial = multivariate_laurentpoly<T>;
using monomial = multivariate_laurent_monomial;
int prime;
unsigned index;
polynomial prepare_polynomial(const polynomial& pol) const {
return (pol - invert_variable(pol, index));
}
bool reduce(const polynomial& pol) const;
public:
periodic_congruence_checker(int pprime = 5,
unsigned ind = 2) :
prime(pprime),
index(ind)
{}
~periodic_congruence_checker() {};
bool operator() (const polynomial& pol) const {
return reduce(prepare_polynomial(pol));
}
};
template<class T>
bool periodic_congruence_checker<T>::reduce(const multivariate_laurentpoly<T>& pol) const {
polynomial res;
for(typename map<monomial, T>::const_iter i = pol.coeffs; i; i++) {
int c = i.key().m[index] % (2 * prime);
if(c < 0)
c += (2 * prime);
monomial mon = monomial(VARIABLE, index, c);
res += polynomial(i.val(), mon);
}
if(verbose)
std::cout << "reduced = " << res << "\n";
return res == 0;
}
template<unsigned p>
class Przytycki_periodicity_checker {
using polynomial = multivariate_laurentpoly<Zp<p>>;
using monomial = multivariate_laurent_monomial;
periodic_congruence_checker<Zp<p>> cong_checker;
public:
Przytycki_periodicity_checker() : cong_checker(p) {}
~Przytycki_periodicity_checker() {}
std::string operator() (const polynomial& pol) const {
std::ostringstream out;
out << knot << ": period = " << p
<< ": "
<< (cong_checker(pol) ? "Maybe" : "No");
return out.str();
}
};
class Kh_periodicity_checker {
using polynomial = multivariate_laurentpoly<Z>;
using monomial = multivariate_laurent_monomial;
unsigned ev_index;
unsigned index;
polynomial khp, leep, quot;
polynomial mul;
void compute_knot_polynomials(knot_diagram& kd);
void compute_quot();
std::pair<polynomial, polynomial> compute_quotient_and_remainder(const polynomial& p,
int period) const;
std::list<polynomial> generate_candidates(const polynomial& q) const;
bool check(const polynomial& q, const polynomial& r, int period) const;
public:
Kh_periodicity_checker(knot_diagram& kd) {
ev_index = 1;
index = 2;
mul = polynomial(Z(1))
+ polynomial(1, VARIABLE, ev_index) *
polynomial(1, VARIABLE, index, 2);
compute_knot_polynomials(kd);
compute_quot();
}
~Kh_periodicity_checker() {}
std::string operator () (int period) const;
};
#endif // _KNOTKIT_PERIODICITY_H

8
planar_diagram.h
View File

@ -1,7 +1,11 @@
#ifndef _KNOTKIT_PLANAR_DIAGRAM_H
#define _KNOTKIT_PLANAR_DIAGRAM_H
/* Planar diagram of a knot. For details, see:
http://katlas.org/wiki/Planar_Diagrams */
#include <lib/vector.h>
#include <string>
class planar_diagram
{
public:
@ -27,3 +31,5 @@ public:
void show_self () const { printf ("planar_diagram %s", name.c_str ()); }
void display_self () const;
};
#endif // _KNOTKIT_PLANAR_DIAGRAM_H

4
simplify_chain_complex.h
View File

@ -1,3 +1,5 @@
#ifndef _KNOTKIT_SIMPLIFY_CHAIN_COMPLEX_H
#define _KNOTKIT_SIMPLIFY_CHAIN_COMPLEX_H
template<class R> class simplified_complex_generators
{
@ -247,3 +249,5 @@ chain_complex_simplifier<R>::chain_complex_simplifier (ptr<const module<R> > C_,
assert (new_d.compose (pi) == pi.compose (d));
assert (pi.compose (iota) == mod_map<R> (new_C, 1));
}
#endif // _KNOTKIT_SIMPLIFY_CHAIN_COMPLEX_H