2016-11-19 21:01:10 +01:00
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#include <periodicity.h>
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#include <simplify_chain_complex.h>
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2016-11-28 09:26:41 +01:00
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bool Przytycki_periodicity_checker::check(int period) const {
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switch(period) {
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case 5: {
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periodic_congruence_checker<Zp<5>> pcc(5);
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return pcc(jones_pol);
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}
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case 7: {
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periodic_congruence_checker<Zp<7>> pcc(7);
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return pcc(jones_pol);
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}
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case 11: {
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periodic_congruence_checker<Zp<11>> pcc(11);
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return pcc(jones_pol);
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}
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case 13: {
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periodic_congruence_checker<Zp<13>> pcc(13);
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return pcc(jones_pol);
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}
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case 17: {
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periodic_congruence_checker<Zp<17>> pcc(17);
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return pcc(jones_pol);
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}
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case 19: {
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periodic_congruence_checker<Zp<19>> pcc(19);
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return pcc(jones_pol);
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}
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}
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}
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std::string Przytycki_periodicity_checker::operator () (int period) const {
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std::ostringstream res;
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res << knot << ": period = " << period << ": "
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<< (check(period) ? "Maybe" : "No");
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return res.str();
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}
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template<class T>
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polynomial_iterator<T>::polynomial_iterator(const multivariate_laurentpoly<T>& pol,
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start_pos sp) {
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for(typename map<monomial, T>::const_iter i = pol.coeffs; i; ++i) {
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monomials.push_back(i.key());
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bounds.push_back(i.val());
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current_pos.push_back(Z(0));
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}
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if(sp == start_pos::begin) {
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level = 0;
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}
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else {
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level = bounds.size();
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}
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#ifndef NDEBUG
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check_current_pos();
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#endif
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}
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#ifndef NDEBUG
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template<class T>
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void polynomial_iterator<T>::check_current_pos() {
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assert(bounds.size() == monomials.size());
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assert(bounds.size() == current_pos.size());
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assert(level <= current_pos.size());
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for(unsigned i = 0; i < current_pos.size(); i++) {
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if(i < level) {
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assert((current_pos[i] <= bounds[i]) &&
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Z(0) <= (current_pos[i]));
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}
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else if(i == level) {
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if(level > 0)
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assert(current_pos[i] <= bounds[i] && Z(0) < current_pos[i]);
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else
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assert((current_pos[i] <= bounds[i]) && Z(0) <= (current_pos[i]));
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}
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else
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assert(current_pos[i] == Z(0));
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}
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}
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#endif // NDEBUG
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template<class T>
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polynomial_iterator<T>&
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polynomial_iterator<T>::operator ++ () {
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#ifndef NDEBUG
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check_current_pos();
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#endif
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if(level == monomials.size())
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return *this;
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unsigned i = 0;
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while(i <= level) {
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#ifndef NDEBUG
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check_current_pos();
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#endif
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if(current_pos[i] < bounds[i]) {
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current_pos[i] += 1;
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break;
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}
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else {
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if(i == level) {
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if(level < monomials.size() - 1) {
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current_pos[i] = 0;
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current_pos[i+1] += 1;
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level++;
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break;
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}
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else {
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level++;
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break;
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}
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}
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else {
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current_pos[i] = 0;
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}
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}
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i++;
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}
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return *this;
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}
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template<class T>
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multivariate_laurentpoly<T> polynomial_iterator<T>::operator *() const {
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polynomial res;
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for(unsigned i = 0; i <= level; i++) {
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res += polynomial(current_pos[i], monomials[i]);
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}
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return res;
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}
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template<class T>
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std::string polynomial_iterator<T>::write_self() const {
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std::ostringstream res;
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res << "level = " << level << std::endl
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<< "monomials:" << std::endl;
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for(auto& mon : monomials)
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res << mon << std::endl;
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res << "bounds: " << std::endl;
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for(auto& b : bounds)
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res << b << std::endl;
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res << "current_pos: " << std::endl;
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for(auto& pos : current_pos)
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res << pos << std::endl;
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return res.str();
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}
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2016-11-19 21:01:10 +01:00
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void Kh_periodicity_checker::compute_knot_polynomials(knot_diagram& kd) {
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unsigned m = kd.num_components ();
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if (m != 1) {
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std::cerr << "warning: this implementation of the criterion works for knots only...";
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exit (EXIT_FAILURE);
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}
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cube<Z2> c (kd, 0);
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ptr<const module<Z2> > C = c.khC;
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mod_map<Z2> d = c.compute_d (1, 0, 0, 0, 0);
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for (unsigned i = 1; i <= kd.n_crossings; i ++)
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d = d + c.H_i (i);
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assert (d.compose (d) == 0);
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// computing Khovanov homology
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if(verbose)
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std::cout << "Computing Khovanov homology" << std::endl;
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{
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chain_complex_simplifier<Z2> s (C, d, maybe<int>(1), maybe<int>(0));
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C = s.new_C;
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d = s.new_d;
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khp = C->free_poincare_polynomial();
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if(verbose)
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std::cout << "KhP = " << khp << "\n";
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}
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// computing Lee homolgy
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if(verbose)
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std::cout << "Computing Lee homology" << std::endl;
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{
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chain_complex_simplifier<Z2> s(C, d, maybe<int>(1), maybe<int>(2));
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C = s.new_C;
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d = s.new_d;
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leep = C->free_poincare_polynomial();
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if(d != 0) {
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std::cout << "For now, you can only use this criterion on Kh-thin knots." << std::endl;
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exit(EXIT_FAILURE);
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}
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if(verbose) {
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std::cout << "LeeP = " << leep << "\n";
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}
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}
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}
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void Kh_periodicity_checker::compute_quot() {
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polynomial diff = khp - leep;
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while(diff != 0) {
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pair<monomial, Z> m = diff.head();
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if(m.first.m[1] == 1) {
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pair<monomial, Z> m1 = diff.tail();
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while(m1.first.m.card() == 1 && m1.first.m[2]) {
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quot += polynomial(m1.second, m1.first);
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polynomial p = polynomial(m1.second, m1.first) * mul;
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diff -= p;
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2016-11-28 09:26:41 +01:00
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if(diff != 0)
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m1 = diff.tail();
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else break;
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2016-11-19 21:01:10 +01:00
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}
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2016-11-28 09:26:41 +01:00
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if(diff != 0)
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m = diff.head();
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else
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break;
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2016-11-19 21:01:10 +01:00
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}
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quot += polynomial(m.second, m.first);
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polynomial p = polynomial(m.second, m.first) * mul;
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diff -= p;
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}
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}
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std::pair<multivariate_laurentpoly<Z>, multivariate_laurentpoly<Z>>
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Kh_periodicity_checker::compute_quotient_and_remainder(const polynomial& quot,
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int period) const {
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polynomial quotient, remainder;
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for(map<monomial, Z>::const_iter i = quot.coeffs; i; i++) {
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std::tuple<Z,Z> div = i.val().divide_with_remainder(period - 1);
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quotient += polynomial(std::get<0>(div), i.key());
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remainder += polynomial(std::get<1>(div), i.key());
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}
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if(verbose) {
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std::cout << "Decomposition of Khp = " << std::endl
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<< leep << " + ("
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<< mul << ") * ("
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<< remainder;
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if(quotient != 0) {
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std::cout << " + " << (period - 1)
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<< " * (" << quotient
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<< ")";
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}
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std::cout << ")" << std::endl;
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}
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return std::make_pair(quotient, remainder);
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}
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bool Kh_periodicity_checker::check(const polynomial& q,
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const polynomial& r,
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int period) const {
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periodic_congruence_checker<Z> pcc(period);
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polynomial t = leep + mul * r;
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if(q == 0) {
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return pcc(t.evaluate(-1,1));
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}
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2016-11-28 09:26:41 +01:00
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if(verbose)
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std::cout << "Checking congruences...";
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polynomial_iterator<Z> pi(q);
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polynomial_iterator<Z> pi_end(q, polynomial_iterator<Z>::start_pos::end);
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Z count = pi.get_count();
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if(verbose)
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std::cout << count << " candidates..." << std::endl;
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Z step = 1;
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if(count >= 32)
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step = count / 32;
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Z c = 0;
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while(pi != pi_end) {
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//std::cout << "pi: " << std::endl << pi;
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polynomial temp = t + polynomial(period - 1) * mul * (*pi);
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if(pcc(temp.evaluate(-1,1))) {
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std::cout << "Candidates:" << std::endl
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<< "EKhP_1 = " << temp << std::endl
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<< "EKhP_" << (period - 1) << " = "
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<< (polynomial(period - 1) * mul *(r - *pi))
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<< std::endl;
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return true;
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2016-11-19 21:01:10 +01:00
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}
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2016-11-28 09:26:41 +01:00
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++pi;
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c += 1;
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if(verbose && c % step == 0)
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std::cout << c << "/" << count << "..." << std::endl;
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2016-11-19 21:01:10 +01:00
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}
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2016-11-28 09:26:41 +01:00
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return false;
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2016-11-19 21:01:10 +01:00
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}
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std::string Kh_periodicity_checker::operator () (int period) const {
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std::ostringstream out;
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std::pair<polynomial, polynomial> q_r = compute_quotient_and_remainder(quot, period);
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bool res = check(std::get<0>(q_r), std::get<1>(q_r), period);
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out << knot << ": period = " << period << ": "
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<< (res ? "Maybe" : "No");
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return out.str();
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}
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