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knotkit/periodicity.cpp

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#include <periodicity.h>
#include <simplify_chain_complex.h>
#include <algorithm>
#include <utility>
#include <fstream>
std::string periodicity_test = "Przytycki";
int period = 5;
extern multivariate_laurentpoly<Z> compute_jones(const knot_diagram& k, bool reduced = false);
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) {
case 5: {
periodic_congruence_checker<Zp<5>> pcc(5);
return pcc(jones_pol);
}
case 7: {
periodic_congruence_checker<Zp<7>> pcc(7);
return pcc(jones_pol);
}
case 11: {
periodic_congruence_checker<Zp<11>> pcc(11);
return pcc(jones_pol);
}
case 13: {
periodic_congruence_checker<Zp<13>> pcc(13);
return pcc(jones_pol);
}
case 17: {
periodic_congruence_checker<Zp<17>> pcc(17);
return pcc(jones_pol);
}
case 19: {
periodic_congruence_checker<Zp<19>> pcc(19);
return pcc(jones_pol);
}
}
return false;
}
std::string Przytycki_periodicity_checker::operator () (int period) const {
std::ostringstream res;
res << knot_name << ": period = " << period << ": "
<< (check(period) ? "Maybe" : "No");
return res.str();
}
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;
}
template<>
std::vector<multivariate_laurentpoly<Z>>
Kh_periodicity_checker::compute_knot_polynomials<Z2>(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::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;
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++;
}
khp = *lee_ss_polynomials.begin();
leep = *lee_ss_polynomials.rbegin();
if(verbose) {
std::cerr << "KhP = " << khp << "\n";
std::cerr << "LeeP = " << leep << "\n";
}
return lee_ss_polynomials;
}
template<typename R>
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...";
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);
// computing Khovanov homology
if(verbose)
std::cerr << "Computing Khovanov homology" << std::endl;
std::vector<polynomial> lee_ss_polynomials;
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;
if(k % 2 == 0) {
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++;
}
khp = *lee_ss_polynomials.begin();
leep = *lee_ss_polynomials.rbegin();
if(verbose) {
std::cerr << "KhP = " << khp << "\n";
std::cerr << "LeeP = " << leep << "\n";
}
return lee_ss_polynomials;
}
Kh_periodicity_checker::Kh_periodicity_checker(knot_diagram& kd,
std::string knot_n,
std::string f = "Z2") :
knot_name(knot_n), field(f) {
ev_index = 1;
index = 2;
quot = std::vector<polynomial>();
mul = std::vector<polynomial>();
if(field == "Z2")
compute_quot(compute_knot_polynomials<Z2>(kd));
else if(field == "Z3")
compute_quot(compute_knot_polynomials<Zp<3>>(kd));
else if(field == "Z5")
compute_quot(compute_knot_polynomials<Zp<3>>(kd));
else if(field == "Z7")
compute_quot(compute_knot_polynomials<Zp<7>>(kd));
else if(field == "Z11")
compute_quot(compute_knot_polynomials<Zp<11>>(kd));
else if(field == "Q")
compute_quot(compute_knot_polynomials<Q>(kd));
else {
std::cerr << "Sorry, I don't recognize field " << f << ". Exiting..." << "\n";
exit(EXIT_FAILURE);
}
}
void Kh_periodicity_checker::compute_quot(const std::vector<polynomial>& lee_ss_polynomials) {
for(unsigned i = 1; i < lee_ss_polynomials.size(); ++i) {
polynomial diff = lee_ss_polynomials[i-1] - lee_ss_polynomials[i];
polynomial q = 0;
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]) {
q += polynomial(m1.second, m1.first);
polynomial p = polynomial(m1.second, m1.first) * mul[i-1];
diff -= p;
if(diff != 0)
m1 = diff.tail();
else break;
}
if(diff != 0)
m = diff.head();
else
break;
}
q += polynomial(m.second, m.first);
polynomial p = polynomial(m.second, m.first) * mul[i-1];
diff -= p;
}
quot.push_back(q);
}
}
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 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::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::cerr << " + " << (period - 1)
<< "*(" << quotient << ")";
std::cerr << ")";
}
std::cerr << "\n";
}
return decomposed_khp;
}
bounds_vector
Kh_periodicity_checker::compute_bounds(const polynomial_tuple& p_tuple, int period) const {
periodic_congruence_checker<Z> pcc(period);
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) {
exp = i.key().m[ev_index];
for(map<unsigned, int>::const_iter j = i.key().m; j; ++j) {
if(j.key() != ev_index) {
int v = j.val() % (2 * period);
if(v < 0) v += (2 * period);
mon *= monomial(VARIABLE, j.key(), v);
}
}
}
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);
mon *= monomial(VARIABLE, j.key(), v);
}
}
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));
if(bounds_v.count(p_temp)) {
if(v_temp >= 0)
bounds_v[p_temp].second += (v_temp * period);
else
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))));
}
}
}
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 bounds_v;
}
Kh_periodicity_checker::Test_Result
Kh_periodicity_checker::check(const polynomial_tuple& polynomials,
int period) const {
periodic_congruence_checker<Z> pcc(period);
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 Test_Result::NO_NONTRIVIAL_DECOMP;
}
std::string Kh_periodicity_checker::operator () (int period) const {
std::ostringstream out;
if(field == "Z5" && Zp<5>(period) == Zp<5>(0))
return "Period (" + std::to_string(period)
+ ") has to be relatively prime to "
+ "the characteristic of the field ("
+ field + ")...";
else if(field == "Z7" && Zp<7>(period) == Zp<7>(0))
return "Period (" + std::to_string(period)
+ ") has to be relatively prime to "
+ "the characteristic of the field ("
+ field + ")...";
else if(field == "Z11" && Zp<11>(period) == Zp<11>(0))
return "Period (" + std::to_string(period)
+ ") has to be relatively prime to "
+ "the characteristic of the field ("
+ field + ")...";
// first check Przytycki's criterion
Przytycki_periodicity_checker P_pc(evaluate_with_copy<Z>(khp, -1, ev_index), knot_name);
if(!P_pc.check(period)) {
out << knot_name << ": period = " << period << ": No (Przytycki's criterion).";
}
else {
auto q_r = compute_quotient_and_remainder(quot, period);
Test_Result res = check(q_r, period);
out << knot_name << ": period = " << period << ": "
<< (res == Test_Result::MAYBE ? "Maybe" :
(res == Test_Result::NO ? "No" : "No (Nontrivial decomposition) ("
+ field + ")"));
}
return out.str();
}
void check_periodicity(knot_diagram& kd, const std::string knot_name, int period, std::string field) {
if(periodicity_test == "all") {
Kh_periodicity_checker Kh_pc(kd, knot_name, field);
for(auto& p : primes_list) {
std::cout << "Kh criterion: "
<< Kh_pc(p) << std::endl;
}
}
else {
if(period == 2 || period == 3) {
std::cout << "Sorry, the criterion doesn'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 19..." << "\n";
exit(EXIT_FAILURE);
}
if(periodicity_test == "Przytycki") {
Przytycki_periodicity_checker P_pc(compute_jones(kd), knot_name);
std::cout << P_pc(period) << std::endl;
}
else if(periodicity_test == "Kh") {
Kh_periodicity_checker Kh_pc(kd, std::string(knot_name), field);
std::cout << Kh_pc(period) << std::endl;
}
else {
std::cout << "Sorry, I don't recognize this option..." << "\n";
exit(EXIT_FAILURE);
}
}
}
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