Verification of periodicity criterion works for nonthin knots.

This commit is contained in:
Wojciech Politarczyk 2017-01-26 11:33:27 +01:00
parent b153876f41
commit f80ac2ce8e
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GPG Key ID: DA54546CA5F66507
3 changed files with 253 additions and 172 deletions

6
kk.cpp
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@ -346,7 +346,7 @@ void check_periodicity(std::string out_file) {
// ten crossings
int num_cr = 10;
int knot_index = stoi(k.substr(3));
for(int i = knot_index; i < rolfsen_crossing_knots(num_cr); i++) {
for(unsigned i = knot_index; i < rolfsen_crossing_knots(num_cr); i++) {
std::string knot_name = std::to_string(num_cr) + "_" + std::to_string(i);
knot_diagram kd_temp = parse_knot(knot_name.c_str());
kd.marked_edge = 1;
@ -362,7 +362,7 @@ void check_periodicity(std::string out_file) {
int knot_index = stoi(k.substr(3));
char alt = k[2];
bool alternating = (alt == 'a' ? true : false);
for(int i = knot_index; i <= htw_knots(num_cr, alternating); i++) {
for(unsigned i = knot_index; i <= htw_knots(num_cr, alternating); i++) {
std::string knot_name = std::to_string(num_cr) + alt + std::to_string(i);
knot_diagram kd_temp = parse_knot(knot_name.c_str());
kd.marked_edge = 1;
@ -378,7 +378,7 @@ void check_periodicity(std::string out_file) {
// at most nine crossings
int num_cr = stoi(k.substr(0, 1));
int knot_index = stoi(k.substr(2));
for(int i = knot_index; i <= rolfsen_crossing_knots(num_cr); i++) {
for(unsigned i = knot_index; i <= rolfsen_crossing_knots(num_cr); i++) {
std::string knot_name = std::to_string(num_cr) + "_" + std::to_string(i);
knot_diagram kd_temp = parse_knot(knot_name.c_str());
kd.marked_edge = 1;

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@ -1,5 +1,10 @@
#include <periodicity.h>
#include <simplify_chain_complex.h>
#include <algorithm>
using polynomial_tuple = std::vector<std::tuple<multivariate_laurentpoly<Z>, multivariate_laurentpoly<Z>, multivariate_laurentpoly<Z>>>;
using bounds_vector = std::map<multivariate_laurentpoly<Z>, std::pair<Z, Z>>;
bool Przytycki_periodicity_checker::check(int period) const {
switch(period) {
@ -38,8 +43,46 @@ std::string Przytycki_periodicity_checker::operator () (int period) const {
return res.str();
}
void Kh_periodicity_checker::compute_knot_polynomials(knot_diagram& kd) {
bool Kh_bounds_iterator::advance() {
if(level == bv.end())
return false;
for(auto bv_it = bv.begin(); bv_it != level; ++bv_it) {
if(current_state[bv_it->first] < (bv_it->second).second) {
current_state[bv_it->first] += period;
for(auto bv_it_2 = bv.begin(); bv_it_2 != bv_it; ++bv_it_2) {
current_state[bv_it_2->first] = bv_it_2->second.first;
}
return true;
}
}
if(current_state[level->first] < bv[level->first].second) {
current_state[level->first] += period;
for(auto bv_it = bv.begin(); bv_it != level; ++bv_it) {
current_state[bv_it->first] = bv_it->second.first;
}
return true;
}
++level;
if(level == bv.end())
return false;
current_state[level->first] += period;
for(auto bv_it = bv.begin(); bv_it != level; ++bv_it) {
current_state[bv_it->first] = bv_it->second.first;
}
return true;
}
multivariate_laurentpoly<Z> Kh_bounds_iterator::get_polynomial() const {
polynomial p;
for(auto& cs : current_state) {
p += cs.second * cs.first;
}
return p;
}
std::vector<multivariate_laurentpoly<Z>>
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...";
@ -56,43 +99,50 @@ void Kh_periodicity_checker::compute_knot_polynomials(knot_diagram& kd) {
// 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));
std::cerr << "Computing Khovanov homology" << std::endl;
std::vector<polynomial> lee_ss_polynomials;
int k = 0;
for(;;) {
chain_complex_simplifier<Z2> s(C, d, maybe<int>(1), maybe<int>(2*k));
C = s.new_C;
d = s.new_d;
khp = C->free_poincare_polynomial();
if(verbose)
std::cout << "KhP = " << khp << "\n";
lee_ss_polynomials.push_back(C->free_poincare_polynomial());
if(k != 0)
mul.push_back(polynomial(Z(1)) + polynomial(Z(1), VARIABLE, 1, 1) * polynomial(Z(1), VARIABLE, 2, 2 * k));
if(d == 0)
break;
k++;
}
// 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);
}
khp = *lee_ss_polynomials.begin();
leep = *lee_ss_polynomials.rbegin();
if(verbose) {
std::cout << "LeeP = " << leep << "\n";
}
std::cerr << "KhP = " << khp << "\n";
std::cerr << "LeeP = " << leep << "\n";
}
// for(unsigned i = 0; i < lee_ss_polynomials.size(); ++i) {
// std::cerr << "lee_ss_polynomials[" << i << "]= "
// << lee_ss_polynomials[i] << "\n";
// std::cerr << "mul[" << i << "] = " << mul[i] << "\n";
// }
return lee_ss_polynomials;
}
void Kh_periodicity_checker::compute_quot() {
polynomial diff = khp - leep;
void Kh_periodicity_checker::compute_quot(const std::vector<polynomial>& lee_ss_polynomials) {
// quot.push_back(polynomial(Z(0)));
for(unsigned i = 1; i < lee_ss_polynomials.size(); ++i) {
polynomial diff = lee_ss_polynomials[i-1] - lee_ss_polynomials[i];
polynomial q = 0;
// std::cerr << "diff = " << diff << "\n";
// std::cerr << "mul = " << mul[i-1] << "\n";
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;
q += polynomial(m1.second, m1.first);
polynomial p = polynomial(m1.second, m1.first) * mul[i-1];
diff -= p;
if(diff != 0)
m1 = diff.tail();
@ -103,42 +153,55 @@ void Kh_periodicity_checker::compute_quot() {
else
break;
}
quot += polynomial(m.second, m.first);
polynomial p = polynomial(m.second, m.first) * mul;
q += polynomial(m.second, m.first);
polynomial p = polynomial(m.second, m.first) * mul[i-1];
diff -= p;
}
quot.push_back(q);
}
// for(unsigned i = 0; i < quot.size(); ++i) {
// std::cerr << "quot[" << i << "] = " << quot[i] << "\n";
// }
}
std::pair<multivariate_laurentpoly<Z>, multivariate_laurentpoly<Z>>
Kh_periodicity_checker::compute_quotient_and_remainder(const polynomial& quot,
int period) const {
polynomial_tuple
Kh_periodicity_checker::compute_quotient_and_remainder(const std::vector<polynomial>& quot, int period) const {
polynomial_tuple decomposed_khp;
for(unsigned i = 0; i < quot.size(); ++i) {
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());
for(map<monomial, Z>::const_iter j = quot[i].coeffs; j; j++) {
std::tuple<Z,Z> div = j.val().divide_with_remainder(period - 1);
quotient += polynomial(std::get<0>(div), j.key());
remainder += polynomial(std::get<1>(div), j.key());
}
decomposed_khp.push_back(std::make_tuple(quotient, remainder, std::move(mul[i])));
}
if(verbose) {
std::cout << "Decomposition of Khp = " << std::endl
<< leep << " + ("
<< mul << ") * ("
std::cerr << "Decomposition of Khp = " << std::endl
<< leep;
for(auto& p: decomposed_khp) {
polynomial quotient, remainder, mul;
tie(quotient, remainder, mul) = p;
std::cerr << " + (" << mul << ") * ("
<< remainder;
if(quotient != 0) {
std::cout << " + " << (period - 1)
<< " * (" << quotient
<< ")";
if(quotient != 0)
std::cerr << " + " << (period - 1)
<< "*(" << quotient << ")";
std::cerr << ")";
}
std::cout << ")" << std::endl;
std::cerr << "\n";
}
return std::make_pair(quotient, remainder);
return decomposed_khp;
}
std::map<multivariate_laurentpoly<Z>, std::pair<Z,Z>>
Kh_periodicity_checker::compute_bounds(const polynomial& p, int period) const {
std::map<polynomial, std::pair<Z, Z>> bounds;
bounds_vector
Kh_periodicity_checker::compute_bounds(const polynomial_tuple& p_tuple, int period) const {
periodic_congruence_checker<Z> pcc(period);
for(map<monomial, Z>::const_iter i = p.coeffs; i; ++i) {
bounds_vector bounds_v;
for(auto& p: p_tuple) {
polynomial quotient, remainder, mul;
tie(quotient, remainder, mul) = p;
for(map<monomial, Z>::const_iter i = quotient.coeffs; i; ++i) {
monomial mon;
int exp = 0;
if(i.key().m % ev_index) {
@ -153,90 +216,84 @@ Kh_periodicity_checker::compute_bounds(const polynomial& p, int period) const {
}
else {
for(map<unsigned, int>::const_iter j = i.key().m; j; ++j) {
int v = j.val() % (2 + period);
if (v < 0) v += (2 * period);
int v = j.val() % (2 * period);
if(v < 0) v += (2 * period);
mon *= monomial(VARIABLE, j.key(), v);
}
}
// std::cout << polynomial(i.val() * pow(-1, exp), mon) << "\n";
// std::cerr << polynomial(i.val() * pow(-1,exp), mon) << "\n";
Z v_temp = i.val() * pow(-1, exp);
polynomial p_temp = (polynomial(1, mon) * mul).evaluate(-1, ev_index);
p_temp = pcc.reduce(p_temp - invert_variable(p_temp, index));
// std::cerr << "p_temp = " << p_temp << "\n";
// std::cerr << "v_temp = " << v_temp << "\n";
// std::cerr << "min_exp = " << min_exp << "\n";
if(bounds_v.count(p_temp)) {
if(v_temp >= 0)
bounds[p_temp].second += (v_temp * period);
bounds_v[p_temp].second += (v_temp * period);
else
bounds[p_temp].first += (v_temp * period);
bounds_v[p_temp].first += (v_temp * period);
}
else if(bounds_v.count(p_temp)) {
if(v_temp >= 0)
bounds_v[p_temp].first -= (v_temp * period);
else
bounds_v[p_temp].second -= (v_temp * period);
}
else {
bounds_v.emplace(p_temp,
std::make_pair<Z,Z>((v_temp < 0 ? (v_temp * period) : Z(0)), (v_temp >= 0 ? (v_temp * period) : Z(0))));
}
}
}
// for(std::map<polynomial, std::pair<Z,Z>>::iterator mi = bounds.begin(); mi != bounds.end(); ++mi) {
// std::cout << "Monomial: " << mi->first << "\n";
// std::cout << "Max: " << std::get<1>(mi->second)
// << ", Min: " << std::get<0>(mi->second) << "\n";
// }
return bounds;
}
std::vector<multivariate_laurentpoly<Z>>
Kh_periodicity_checker::compute_basis_polynomials(int period) const {
std::vector<polynomial> res;
periodic_congruence_checker<Z> pcc(period);
for(int i = 1; i < period; i += 2) {
res.push_back(pcc.reduce(get_basis_polynomial(i)));
if(verbose) {
for(auto& t: bounds_v) {
Z neg, pos;
tie(neg, pos) = t.second;
std::cerr << "polynomial = " << t.first << "\n";
std::cerr << "min = " << neg << ", max = " << pos << "\n";
}
return res;
}
return bounds_v;
}
multivariate_laurentpoly<Z> Kh_periodicity_checker::get_basis_polynomial(monomial mon) const {
return (polynomial(Z(1), mon) * mul).evaluate(-1, ev_index) -
invert_variable((polynomial(Z(1), mon) * mul).evaluate(-1, ev_index), index);
}
bool Kh_periodicity_checker::check(const polynomial& q,
const polynomial& r,
Test_Result Kh_periodicity_checker::check(const polynomial_tuple& polynomials,
int period) const {
periodic_congruence_checker<Z> pcc(period);
polynomial t = (leep + mul * (r - q)).evaluate(-1, ev_index);
t = pcc.reduce(t - invert_variable(t, index));
if(pcc(t)) {
return true;
}
else if(q == 0)
return false;
// std::cout << t << std::endl;
// std::cout << q << "\n";
std::map<polynomial, std::pair<Z,Z>> bounds = compute_bounds(q, period);
for(std::map<polynomial, std::pair<Z,Z>>::iterator it = bounds.begin();
it != bounds.end(); ++it) {
polynomial mon = it->first;
}
std::vector<polynomial> basis_polynomials = compute_basis_polynomials(period);
polynomial p = pcc.reduce(get_basis_polynomial(2 * period - 1));
for(Z i = bounds[p].first; i <= bounds[p].second; i += 5) {
polynomial p_temp = t + polynomial(i, VARIABLE, index, 0) * p;
// std::cout << "i = " << i << "\n";
// std::cout << "p_temp = " << p_temp << "\n";
if(p_temp == 0)
return true;
for(std::vector<polynomial>::iterator it = basis_polynomials.begin(); it != basis_polynomials.end(); ++it) {
pair<monomial, Z> m = p_temp.coeffs.head();
monomial mon = m.first;
Z c = m.second;
polynomial pp = pcc.reduce(get_basis_polynomial(mon));
if(pp == *it) {
if(c < bounds[pp].first || c > bounds[pp].second)
break;
else {
// std::cout << "pp = " << pp << "\n";
p_temp -= polynomial(c, VARIABLE, index, 0) * pp;
// std::cout << "p_temp = " << p_temp << "\n";
if(p_temp == 0)
return true;
}
polynomial t = polynomial(COPY, leep);
for(auto& p : polynomials) {
polynomial quotient, remainder, mul;
tie(quotient, remainder, mul) = p;
t += mul * (remainder - quotient);
//std::cerr << "t = " << t << "\n";
}
polynomial s = t.evaluate(-1, ev_index);
s = pcc.reduce(s - invert_variable(s, index));
if(pcc(s)) {
return Test_Result::MAYBE;
}
else if(all_of(polynomials.begin(), polynomials.end(),
[](std::tuple<polynomial, polynomial, polynomial> t)
{ return get<0>(t) == 0; }))
return Test_Result::NO;
bounds_vector bounds = compute_bounds(polynomials, period);
if(verbose)
std::cerr << "s = " << s << "\n";
Kh_bounds_iterator Kh_b_it(bounds, period);
if(verbose)
std::cerr << "current_state = " << Kh_b_it.get_polynomial() << "\n";
if(Kh_b_it.get_polynomial() == s)
return Test_Result::MAYBE;
while(Kh_b_it.advance()) {
if(verbose)
std::cerr << "current_state = " << Kh_b_it.get_polynomial() << "\n";
if(s == Kh_b_it.get_polynomial())
return Test_Result::MAYBE;
}
return false;
return Test_Result::NO_NONTRIVIAL_DECOMP;
}
std::string Kh_periodicity_checker::operator () (int period) const {
@ -247,10 +304,11 @@ std::string Kh_periodicity_checker::operator () (int period) const {
out << knot_name << ": period = " << period << ": No (Przytycki's criterion).";
}
else {
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);
auto q_r = compute_quotient_and_remainder(quot, period);
Test_Result res = check(q_r, period);
out << knot_name << ": period = " << period << ": "
<< (res ? "Maybe" : "No");
<< (res == Test_Result::MAYBE ? "Maybe" :
(res == Test_Result::NO ? "No" : "No (Nontrivial decomposition)."));
}
return out.str();
}

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@ -6,6 +6,7 @@
#include <string>
#include <vector>
#include <utility>
#include <tuple>
extern bool verbose;
extern const char* knot;
@ -14,6 +15,8 @@ extern std::string periodicity_test;
const std::vector<int> primes_list = {5, 7, 11, 13, 17, 19};
enum class Test_Result { MAYBE, NO, NO_NONTRIVIAL_DECOMP };
const unsigned eval_index = 1;
const unsigned invert_index = 2;
@ -59,6 +62,8 @@ periodic_congruence_checker<T>::reduce(const multivariate_laurentpoly<T>& pol) c
monomial mon = monomial(VARIABLE, index, c);
res += polynomial(i.val(), mon);
}
// if(verbose)
// std::cout << "res = " << res << "\n";
return res;
}
@ -78,43 +83,61 @@ class Przytycki_periodicity_checker {
std::string operator() (int period) const;
};
class Kh_bounds_iterator {
using polynomial = multivariate_laurentpoly<Z>;
using monomial = multivariate_laurent_monomial;
using polynomial_tuple = std::vector<std::tuple<polynomial, polynomial, polynomial>>;
using bounds_vector = std::map<multivariate_laurentpoly<Z>, std::pair<Z, Z>>;
bounds_vector bv;
int period;
std::map<polynomial, Z> current_state;
std::map<polynomial, std::pair<Z,Z>>::iterator level;
public:
Kh_bounds_iterator(bounds_vector v, int p) :
bv(v), period(p) {
for(auto& v: bv) {
current_state[v.first] = v.second.first;
}
level = bv.begin();
}
~Kh_bounds_iterator() {}
bool advance();
polynomial get_polynomial() const;
};
class Kh_periodicity_checker {
using polynomial = multivariate_laurentpoly<Z>;
using monomial = multivariate_laurent_monomial;
using polynomial_tuple = std::vector<std::tuple<polynomial, polynomial, polynomial>>;
using bounds_vector = std::map<multivariate_laurentpoly<Z>, std::pair<Z, Z>>;
unsigned ev_index;
unsigned index;
polynomial khp, leep, quot;
polynomial mul;
polynomial khp, leep;
std::vector<polynomial> quot, mul, quotients, remainders;
std::string knot_name;
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::map<polynomial, std::pair<Z,Z>>
compute_bounds(const polynomial& p, int period) const;
polynomial get_basis_polynomial(int exp) const {
return (polynomial(1, VARIABLE, index, exp) * mul).evaluate(-1, ev_index) -
invert_variable((polynomial(1, VARIABLE, index, exp) * mul).evaluate(-1, ev_index), index);
}
polynomial get_basis_polynomial(monomial mon) const;
std::vector<polynomial> compute_basis_polynomials(int period) const;
bool check(const polynomial& q, const polynomial& r, int period) const;
std::vector<polynomial> compute_knot_polynomials(knot_diagram& kd);
void compute_quot(const std::vector<polynomial>& lee_ss_polynomials);
polynomial_tuple
compute_quotient_and_remainder(const std::vector<polynomial>& p, int period) const;
bounds_vector
compute_bounds(const polynomial_tuple& p, int period) const;
Test_Result check(const polynomial_tuple& polynomials, int period) const;
public:
Kh_periodicity_checker(knot_diagram& kd, std::string knot_n) :
knot_name(knot_n) {
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();
quot = std::vector<polynomial>();
mul = std::vector<polynomial>();
compute_quot(compute_knot_polynomials(kd));
}
~Kh_periodicity_checker() {}