547 lines
19 KiB
Python
547 lines
19 KiB
Python
#!/usr/bin/env python
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# Copyright (c) 2018: Maria Marchwicka, Wojciech Politarczyk.
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# This program is free software: you can redistribute it and/or modify
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# it under the terms of the GNU General Public License as published by
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# the Free Software Foundation, either version 3 of the License, or
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# (at your option) any later version.
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#
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# This program is distributed in the hope that it will be useful,
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# but WITHOUT ANY WARRANTY; without even the implied warranty of
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# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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# GNU General Public License for more details.
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#
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# You should have received a copy of the GNU General Public License
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# along with this program. If not, see <http://www.gnu.org/licenses/>
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import sys
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import os
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import numpy as np
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import warnings
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class MySettings(object):
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def __init__(self):
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self.f_pd_knot_11_15 = os.path.join(os.getcwd(), "knots_11_15.txt")
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self.f_knot_up_to_10 = os.path.join(os.getcwd(), "knots_3_10.txt")
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self.f_homfly_lm_in = os.path.join(os.getcwd(), "homflypt.input")
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self.f_results_out = os.path.join(os.getcwd(), "results.out")
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self.periods = [3, 5, 7, 9, 11]
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self.set_to_check = self.get_set()
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# check only knots from defined set
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self.only_chosen = True
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self.only_chosen = False
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self.print_results = False
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self.print_results = True
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# HOMFLYPT polynomials from file
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self.input_file_with_homflypt = True
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# self.input_file_with_homflypt = False
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if self.input_file_with_homflypt:
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if not os.path.isfile(self.f_homfly_lm_in):
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warnings.warn("No input file with HOMFLYPT polynomials")
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self.input_file_with_homflypt = False
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def get_set(self):
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set_to_check = set()
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return set_to_check
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class PeriodicityTester(object):
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def __init__(self, name, pd_code, A=None, f_homfly_in=None):
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self.results = []
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'''
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To results for each period q a list in following form will be appended:
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[q, murasugi, naik_1, naik_2, borodzik, przytycki]
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Crierion is set to be:
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-1 if it is not applicable (details in check_naik_2, check_przytycki,
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1 if criterion doesn't exclude periodic,
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0 if criterion excludes periodicity.
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murasugi, naik_1, naik_2 or borodzik is also set to be:
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2 if alexander_polynomial == 1.
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0 if previous criterion in the list is 0.
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'''
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self.name = name
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self.pd_code = pd_code
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self.smith = None
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self.reset_results()
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if pd_code is not None:
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self.K = Link(pd_code)
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self.seifert = self.K.seifert_matrix()
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else:
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self.seifert = A
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# delta := Alexander polynomial
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delta = (self.seifert.transpose() - t * self.seifert).determinant()
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self.delta = delta.shift(-delta.exponents()[0])
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self.delta_factors = self.set_delta_factors()
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self.przytycki_tester = self.get_przytycki_tester(f_homfly_in)
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def reset_results(self):
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self.murasugi = 0
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self.naik_1 = 0
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self.naik_2 = 0
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self.borodzik = 0
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self.przytycki = 0
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self.murasugi_fulfilling = set()
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self.naik_1_fulfilling = []
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self.naik_2_fulfilling = []
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def set_smith(self):
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symetric_from_seifert = self.seifert + self.seifert.transpose()
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assert symetric_from_seifert.determinant() != 0, \
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"The determinant of A + A^T is zero."
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self.smith = symetric_from_seifert.smith_form()
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D, U, V = self.smith
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self.diagonal = D.diagonal()
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self.maximum_in_diagonal = max(self.diagonal)
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C = U.inverse()
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E_inverse = V
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self.C_tran_E_inv_D_inv = C.transpose() * E_inverse * D.inverse()
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self.matrix_C = C
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self.matrix_E_inverse = E_inverse
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def get_przytycki_tester(self, f_homfly_in):
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if self.pd_code is not None:
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try:
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return PrzytyckiTester(self.K, self.name, f_homfly_in)
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except ImportError as e:
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pass
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return None
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def get_C_tran_E_inv_D_inv(self):
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if self.smith is None:
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self.set_smith()
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return self.C_tran_E_inv_D_inv
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def get_maximum_in_diagonal(self):
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if self.smith is None:
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self.set_smith()
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return self.maximum_in_diagonal
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def set_delta_factors(self):
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# find all delta (alexander polynomial) factors
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lst_of_factors = [[f[0]] * f[1] for f in self.delta.factor()]
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# flattening a list
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lst_of_factors = [el for sublist in lst_of_factors for el in sublist]
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delta_candidates = set()
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for s in get_subsets(lst_of_factors):
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d = t^0
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for el in s:
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d *= el
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delta_candidates.add(d)
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return delta_candidates
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def check_criteria_for_period(self, q):
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self.reset_results()
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self.przytycki = self.check_przytycki(q)
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if self.delta == 1:
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self.murasugi = 2
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self.naik_1 = 2
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self.naik_2 = 2
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self.borodzik = 2
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return 2
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self.murasugi = self.check_murasugi(q)
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self.naik_1 = self.check_naik_1(q)
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self.naik_2 = self.check_naik_2(q)
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self.borodzik = self.check_borodzik(q)
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return self.borodzik * self.przytycki
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def check_murasugi(self, q):
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'''
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Select these delta factors and natural number r such that:
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delta = delta_prime^q * (1 + t^1 + ... + t^(r-1))^(q-1) mod q
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where "delta_prime" is a delta factor.
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'''
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quotient_delta = self.delta.change_ring(GF(q))
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# Underlying polynomial of quotient_delta:
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quotient_delta = quotient_delta.polynomial_construction()[0]
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delta_degree = quotient_delta.degree()
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for candidate in self.delta_factors:
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quotient_candidate = candidate.change_ring(GF(q))
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power_candidate = quotient_candidate^q
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power_candidate = power_candidate.polynomial_construction()[0]
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# (r - 1) - possible t-polynomial degree
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r = (delta_degree - power_candidate.degree()) / (q - 1) + 1
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if r < 1 or not r.is_integer():
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continue
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t_polynomial = get_t_polynomial(q, r)
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right_side = t_polynomial * power_candidate
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if quotient_delta != right_side and -quotient_delta != right_side:
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continue
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self.murasugi_fulfilling.add((candidate, r))
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return int(bool(self.murasugi_fulfilling))
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def check_naik_1_candidate(self, delta_prime, delta_evaluated, q):
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t_delta = delta_evaluated / delta_prime(-1)
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t_delta_dict = {f[0]: f[1] for f in factor(t_delta)}
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t_delta_factors = [f for f in t_delta_dict.keys()
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if f != 2 and gcd(q, f) == 1]
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for f in t_delta_factors:
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f_q = naik_number_dict.setdefault((f, q), get_naik_number(f, q))
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if not (t_delta_dict[f] / (2 * f_q)).is_integer():
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return None
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return t_delta_factors
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def check_naik_1(self, q):
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'''
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For each delta' find a set P of prime numbers p such that:
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gcd(p, q) == 1, p != 2 and p| t_delta, t_delta = delta(-1)/delta'(-1).
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Check if all p factors of t_delta has multiplicity divisible by 2*[p|q].
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If it holds for at least one delta' candidate, set naik_1 = True.
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'''
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delta_evaluated = self.delta(-1)
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for delta_prime, _ in self.murasugi_fulfilling:
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t_delta_factors = self.check_naik_1_candidate(delta_prime,
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delta_evaluated, q)
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if t_delta_factors is not None:
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self.naik_1_fulfilling.append((delta_prime, t_delta_factors))
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return int(bool(self.naik_1_fulfilling))
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def check_naik_2_candidate(self, q, p_list):
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delta_prime_bases = []
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maximum_in_diagonal = self.get_maximum_in_diagonal()
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for p in p_list:
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p_q = naik_number_dict[(p, q)]
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bases_for_p_torsion = []
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factor_power = p
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# find all p^k torsion parts
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while (maximum_in_diagonal / factor_power).is_integer():
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basis_for_p_k_part = []
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for el in self.diagonal:
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to_be_append = el / factor_power
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is_int = (to_be_append / p).is_integer()
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if to_be_append.is_integer() and not is_int:
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basis_for_p_k_part.append(to_be_append)
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else:
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basis_for_p_k_part.append(0)
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len_non_zero = sum(x != 0 for x in basis_for_p_k_part)
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# check if dimension is multiple of 2 * naik_number
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if not (len_non_zero / (2 * p_q)).is_integer():
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return None
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factor_power *= p
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bases_for_p_torsion.append(basis_for_p_k_part)
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delta_prime_bases.append((p, bases_for_p_torsion))
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return delta_prime_bases
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def check_naik_2(self, q):
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'''
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For each delta' consider a set P of primes p such that: gcd(p, q) == 1,
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p != 2, p| delta(-1)/delta'(-1) (self.naik_1_fulfilling) and p is not
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a factor of delta'(-1). Check if dimension of p^k torsion part
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is divisible by 2*[p|q] for all k and all p from P.
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If it holds for at least one delta' candidate, we set naik_2 to be True.
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In particular naik_2 is set to be -1 if the criterion passes,
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but only in cases where P is an empty set.
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'''
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for delta_prime, p_list in self.naik_1_fulfilling:
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delta_prime_factors = set([d[0] for d in factor(delta_prime(-1))])
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p_list = [p for p in p_list if p not in delta_prime_factors]
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if not p_list:
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self.naik_2 = -1
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self.borodzik = -1
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continue
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delta_prime_bases = self.check_naik_2_candidate(q, p_list)
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if delta_prime_bases is not None:
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self.naik_2_fulfilling.append((delta_prime,
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delta_prime_bases))
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if self.naik_2_fulfilling:
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return 1
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return self.naik_2
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def check_borodzik(self, q):
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'''
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Consider all delta' that meet criterion Naik 2.
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For all p from a set P (defined as in check_naik_2)
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and all k consider p^k torsion part.
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For each p^k torsion check if eta == epsilon_1 * epsilon_2
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(see check_borodzik_candidate()).
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If it holds for at least one delta' candidate, set borodzik to be True.
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In particular borodzik is set to be -1 if the criterion passes,
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but only in cases where P is an empty set.
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'''
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for delta_prime, delta_prime_bases in self.naik_2_fulfilling:
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borodzik_pass = True
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for p, bases_for_p in delta_prime_bases:
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# if len(bases_for_p) > 1:
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# print "HURA" # more than one p^k part - not found yet
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if not self.check_borodzik_candidate(q, p, bases_for_p):
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borodzik_pass = False
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break
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if borodzik_pass:
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return 1
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return self.borodzik
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def check_borodzik_candidate(self, q, p, bases):
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'''
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For each p^k torsion check if eta == epsilon_1 * epsilon_2.
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If determinant of corsesponding matrix P is square modulo p, then:
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episilon_1 = 1, else: episilon_1 = -1.
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If p == 3 mod(4) and a rank of p^k torsion part n == 2 mod(4), then:
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epsilon_2 = -1, else: epsilon_2 = 1.
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eta = naik_sign ^ d, where d = n / (2 * [p, q]).
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If p^([p, q]) % q == 1, then: naik_sign = 1, else: naik_sign = -1.
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'''
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for k, p_k_basis in enumerate(bases):
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X = np.diagflat(p_k_basis)
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# columns that up to zero (element in diagonal is zero):
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zero_columns = np.nonzero(X.sum(axis=0) == 0)
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X = np.delete(X, zero_columns, axis=1)
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n = X.shape[1]
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X = matrix(X)
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P = p^(k + 1) * X.transpose() * self.get_C_tran_E_inv_D_inv() * X
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P_det = P.determinant()
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if P_det % p == 0:
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raise ValueError("P determinant is 0 modulo p.")
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if p % 4 == 3 and n % 4 == 2: # epsilon_1
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epsilon = -1
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else:
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epsilon = 1
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if not mod(P_det, p).is_square():
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epsilon *= -1 # epsilon = epsilon_1 * epsilon_2
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p_q = naik_number_dict[(p, q)]
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d = n / (2 * p_q)
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# sign(p_q) - whether rest is -1 or 1
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if sign(p_q)^d != epsilon:
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return False
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return True
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def check_przytycki(self, q):
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if self.przytycki_tester is not None and q in prime_numbers:
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try:
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return self.przytycki_tester.check_congruence(q)
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except (AttributeError, OverflowError) as e:
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pass
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return -1
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def save_results(self, f_out):
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for result in self.results:
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line_to_write = self.name + "," + ",".join(map(str, result))
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f_out.writelines(line_to_write + "\n")
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def print_results(self):
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print "\n" + "#" * 15 + " " + str(self.name) + " " + "#" * 15
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for result in self.results:
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q = result[0]
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print
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self.print_przytycki_result(q, result[5])
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if result[1] == 2:
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print "Alexander polynomial is 1"
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continue
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if not result[1]:
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print "\t\tMurasugi: fail, q = " + str(q)
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continue
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print "Murasugi: pass, q = " + str(q)
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if not result[2]:
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print "\t\tNaik 1: fail, q = " + str(q)
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continue
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print "Naik 1: pass, q = " + str(q)
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if not result[3]:
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print "\t\tNaik 2: fail, q = " + str(q)
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continue
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if result[3] == -1:
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print "Naik 2: not applicable, q = " + str(q)
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continue
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print "Naik 2: pass, q = " + str(q)
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if not result[4]:
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print ("\t\tBorodzik: fail, q = " + str(q))
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continue
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if result[4] == -1:
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print ("Borodzik: not applicable, q = " + str(q))
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continue
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print ("Borodzik: pass, q = " + str(q))
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def print_przytycki_result(self, q, result):
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if not result:
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print "\t\tPrzytycki: fail, q = " + str(q)
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elif result == -1:
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print "Przytycki: not applicable, q = " + str(q)
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else:
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print "Przytycki: pass, q = " + str(q)
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class PrzytyckiTester(object):
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def __init__(self, K, name, f_homfly_in=None):
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homflypt = self.get_homflypt_polynomial(K, name, f_homfly_in)
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homfly_difference = homflypt(a, -z) - homflypt(a^-1, -z)
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self.homfly_difference = z * homfly_difference
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self.homflypt_polynomial = homflypt
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def get_homflypt_polynomial(self, K, name, f_homfly_in=None):
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if f_homfly_in is not None:
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try:
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current_name, homflypt = f_homfly_in.readline().split(',')
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while current_name != name:
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current_name, homflypt = f_homfly_in.readline().split(',')
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homflypt = sage_eval(homflypt, locals={'a': a, 'z': z})
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return homflypt
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except (AttributeError, ValueError) as e:
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pass
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return K.homfly_polynomial('a', 'z', 'lm')
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def check_congruence(self, q):
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for i in range(q + 1):
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z_coefficient = self.homfly_difference.coefficient(z^(i+1))
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ideal = (a + a^-1)^(q - i) # for i == q will be 1
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coefficient_modulo_ideal = z_coefficient.quo_rem(ideal)[1]
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coefficient_modulo_q = coefficient_modulo_ideal.change_ring(GF(q))
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if coefficient_modulo_q != 0:
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return 0
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return 1
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def check_criteria(name, pd_code, f_homfly_in=None):
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tester = PeriodicityTester(name, pd_code, None, f_homfly_in)
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for i, q in enumerate(settings.periods):
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tester.check_criteria_for_period(q)
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tester.results.append([q, tester.murasugi, tester.naik_1,
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tester.naik_2, tester.borodzik,
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tester.przytycki])
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if settings.print_results:
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tester.print_results()
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return tester
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def get_naik_number(p, q):
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'''
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Calculate the smallest integer i such that p^i == +/-1 mod q.
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Signum of i shows whether rest is -1 or 1
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'''
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if gcd(q, p) > 1:
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return 0
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p_power = p
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for i in xrange(1, sys.maxint):
|
|
pq = p_power % q
|
|
if pq == 1:
|
|
return i
|
|
if pq == q - 1:
|
|
return -i
|
|
p_power *= p
|
|
|
|
|
|
def get_t_polynomial(q, r): # for check_murasugi(), r coresponds to l in paper
|
|
t_polynomial = sum([t^i for i in range(r)])
|
|
t_polynomial = t_polynomial.change_ring(GF(q))
|
|
t_polynomial ^= (q - 1)
|
|
return t_polynomial
|
|
|
|
|
|
def get_subsets(myset):
|
|
return reduce(lambda z, x: z + [y + [x] for y in z], myset, [[]])
|
|
|
|
|
|
def parse_pd_code(pd_code_from_file):
|
|
set = '0987654321[],'
|
|
pd_code = ''.join([c for c in pd_code_from_file if c in set])
|
|
return eval(pd_code)
|
|
|
|
|
|
def parse_knot_name(name):
|
|
data = name[5: -2].split(',')
|
|
name = data[0].strip() + data[1].strip().lower()[:1] + data[2].strip()
|
|
return name
|
|
|
|
|
|
def check_11_to_15(f_out, f_homfly_out=None, f_homfly_in=None):
|
|
with open(settings.f_pd_knot_11_15, 'r') as f:
|
|
line = f.readline()
|
|
while line:
|
|
name = parse_knot_name(line)
|
|
pd_code = parse_pd_code(f.readline())
|
|
line = f.readline()
|
|
tester = check_criteria(name, pd_code, f_homfly_in)
|
|
if tester is None:
|
|
continue
|
|
tester.save_results(f_out)
|
|
|
|
|
|
def check_up_to_10(f_out, f_homfly_in=None):
|
|
with open(settings.f_knot_up_to_10, 'r') as f:
|
|
line = f.readline()
|
|
while line:
|
|
line = line.split(" = ")
|
|
name = str(line[0])[5:]
|
|
pd_code = parse_pd_code(str(line[1]))
|
|
line = f.readline()
|
|
tester = check_criteria(name, pd_code, f_homfly_in)
|
|
if tester is None:
|
|
continue
|
|
tester.save_results(f_out)
|
|
|
|
|
|
def test_all(f_out, f_homfly_in=None):
|
|
check_up_to_10(f_out, f_homfly_in)
|
|
if f_out is not None:
|
|
f_out.flush()
|
|
check_11_to_15(f_out, f_homfly_in)
|
|
|
|
|
|
if __name__ == '__main__':
|
|
|
|
settings = MySettings()
|
|
S.<a, z> = LaurentPolynomialRing(ZZ)
|
|
R.<t> = LaurentPolynomialRing(ZZ)
|
|
prime_numbers = Primes()
|
|
naik_number_dict = {}
|
|
if settings.input_file_with_homflypt:
|
|
with open(settings.f_results_out, 'w') as f_out,\
|
|
open(settings.f_homfly_lm_in, 'r') as f_homfly_in:
|
|
test_all(f_out, f_homfly_in)
|
|
else:
|
|
with open(settings.f_results_out, 'w') as f_out:
|
|
test_all(f_out)
|