python 3, refac
This commit is contained in:
Maria Marchwicka
parent
42eb3a6364
commit
c5c788e89c
118
periodicity.sage
118
periodicity.sage
@ -1,3 +1,5 @@
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#!/usr/bin/python
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# Copyright (c) 2018: Maria Marchwicka, Wojciech Politarczyk.
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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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# 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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# it under the terms of the GNU General Public License as published by
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@ -31,11 +33,6 @@ class MySettings(object):
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self.f_results_out = os.path.join(os.getcwd(), "results.out")
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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.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 = False
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@ -51,17 +48,15 @@ class MySettings(object):
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warnings.warn("No input file with HOMFLYPT polynomials")
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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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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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class PeriodicityTester(object):
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def __init__(self, name, pd_code, A=None, f_homfly_in=None):
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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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'''
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To results for each period q a list in following form will be appended:
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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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[q, murasugi, naik_1, naik_2, borodzik, przytycki]
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@ -74,6 +69,7 @@ class PeriodicityTester(object):
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0 if previous criterion in the list is 0.
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0 if previous criterion in the list is 0.
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'''
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'''
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self.results = []
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self.name = name
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self.name = name
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self.pd_code = pd_code
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self.pd_code = pd_code
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@ -85,7 +81,7 @@ class PeriodicityTester(object):
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self.seifert = self.K.seifert_matrix()
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self.seifert = self.K.seifert_matrix()
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else:
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else:
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self.seifert = A
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self.seifert = A
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# delta := Alexander polynomial
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# delta is the normalized Alexander polynomial
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delta = (self.seifert.transpose() - t * self.seifert).determinant()
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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 = delta.shift(-delta.exponents()[0])
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self.delta_factors = self.set_delta_factors()
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self.delta_factors = self.set_delta_factors()
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@ -135,6 +131,8 @@ class PeriodicityTester(object):
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def set_delta_factors(self):
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def set_delta_factors(self):
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# find all delta (alexander polynomial) factors
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# find all delta (alexander polynomial) factors
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if self.delta == 1:
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return [1]
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lst_of_factors = [[f[0]] * f[1] for f in self.delta.factor()]
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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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# 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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lst_of_factors = [el for sublist in lst_of_factors for el in sublist]
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@ -199,8 +197,8 @@ class PeriodicityTester(object):
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t_delta_factors = [f for f in t_delta_dict.keys()
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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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if f != 2 and gcd(q, f) == 1]
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for f in t_delta_factors:
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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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q_f = naik_number_dict.setdefault((q, f), get_naik_number(q, f))
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if not (t_delta_dict[f] / (2 * f_q)).is_integer():
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if not (t_delta_dict[f] / (2 * q_f)).is_integer():
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return None
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return None
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return t_delta_factors
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return t_delta_factors
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@ -208,7 +206,7 @@ class PeriodicityTester(object):
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'''
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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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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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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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Check if all p factors of t_delta has multiplicity divisible by 2*[q|p].
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If it holds for at least one delta' candidate, set naik_1 = True.
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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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'''
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delta_evaluated = self.delta(-1)
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delta_evaluated = self.delta(-1)
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@ -225,7 +223,7 @@ class PeriodicityTester(object):
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delta_prime_bases = []
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delta_prime_bases = []
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maximum_in_diagonal = self.get_maximum_in_diagonal()
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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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for p in p_list:
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p_q = naik_number_dict[(p, q)]
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q_p = naik_number_dict[(q, p)]
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bases_for_p_torsion = []
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bases_for_p_torsion = []
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factor_power = p
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factor_power = p
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# find all p^k torsion parts
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# find all p^k torsion parts
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@ -240,7 +238,7 @@ class PeriodicityTester(object):
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basis_for_p_k_part.append(0)
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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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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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# 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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if not (len_non_zero / (2 * q_p)).is_integer():
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return None
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return None
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factor_power *= p
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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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bases_for_p_torsion.append(basis_for_p_k_part)
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@ -257,6 +255,7 @@ class PeriodicityTester(object):
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In particular naik_2 is set to be -1 if the criterion passes,
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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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but only in cases where P is an empty set.
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'''
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'''
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# Proposition 2.8.
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for delta_prime, p_list in self.naik_1_fulfilling:
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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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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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p_list = [p for p in p_list if p not in delta_prime_factors]
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@ -290,7 +289,7 @@ class PeriodicityTester(object):
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borodzik_pass = True
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borodzik_pass = True
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for p, bases_for_p in delta_prime_bases:
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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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# 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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# 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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if not self.check_borodzik_candidate(q, p, bases_for_p):
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borodzik_pass = False
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borodzik_pass = False
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break
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break
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@ -305,8 +304,8 @@ class PeriodicityTester(object):
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episilon_1 = 1, else: episilon_1 = -1.
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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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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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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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eta = naik_sign ^ d, where d = n / (2 * [q, p]).
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If p^([p, q]) % q == 1, then: naik_sign = 1, else: naik_sign = -1.
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If p^([q, p]) % q == 1, then: naik_sign = 1, else: naik_sign = -1.
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'''
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'''
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for k, p_k_basis in enumerate(bases):
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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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X = np.diagflat(p_k_basis)
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@ -328,10 +327,10 @@ class PeriodicityTester(object):
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if not mod(P_det, p).is_square():
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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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epsilon *= -1 # epsilon = epsilon_1 * epsilon_2
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p_q = naik_number_dict[(p, q)]
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q_p = naik_number_dict[(q, p)]
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d = n / (2 * p_q)
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d = n / (2 * q_p)
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# sign(p_q) - whether rest is -1 or 1
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# sign(q_p) - whether rest is -1 or 1
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if sign(p_q)^d != epsilon:
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if sign(q_p)^d != epsilon:
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return False
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return False
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return True
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return True
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@ -351,57 +350,57 @@ class PeriodicityTester(object):
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def print_results(self):
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def print_results(self):
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print "\n" + "#" * 15 + " " + str(self.name) + " " + "#" * 15
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print("\n" + "#" * 15 + " " + str(self.name) + " " + "#" * 15)
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for result in self.results:
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for result in self.results:
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q = result[0]
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q = result[0]
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print
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print()
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self.print_przytycki_result(q, result[5])
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self.print_przytycki_result(q, result[5])
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if result[1] == 2:
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if result[1] == 2:
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print "Alexander polynomial is 1"
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print("Alexander polynomial is 1")
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continue
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continue
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if not result[1]:
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if not result[1]:
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print "\t\tMurasugi: fail, q = " + str(q)
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print("\t\tMurasugi: fail, q = " + str(q))
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continue
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continue
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print "Murasugi: pass, q = " + str(q)
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print("Murasugi: pass, q = " + str(q))
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if not result[2]:
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if not result[2]:
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print "\t\tNaik 1: fail, q = " + str(q)
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print("\t\tNaik 1: fail, q = " + str(q))
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continue
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continue
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print "Naik 1: pass, q = " + str(q)
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print("Naik 1: pass, q = " + str(q))
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if not result[3]:
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if not result[3]:
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print "\t\tNaik 2: fail, q = " + str(q)
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print("\t\tNaik 2: fail, q = " + str(q))
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continue
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continue
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if result[3] == -1:
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if result[3] == -1:
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print "Naik 2: not applicable, q = " + str(q)
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print("Naik 2: not applicable, q = " + str(q))
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continue
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continue
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print "Naik 2: pass, q = " + str(q)
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print("Naik 2: pass, q = " + str(q))
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if not result[4]:
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if not result[4]:
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print ("\t\tBorodzik: fail, q = " + str(q))
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print("\t\tBorodzik: fail, q = " + str(q))
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continue
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continue
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if result[4] == -1:
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if result[4] == -1:
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print ("Borodzik: not applicable, q = " + str(q))
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print("Borodzik: not applicable, q = " + str(q))
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continue
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continue
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print ("Borodzik: pass, q = " + str(q))
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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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def print_przytycki_result(self, q, result):
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if not result:
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if not result:
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print "\t\tPrzytycki: fail, q = " + str(q)
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print("\t\tPrzytycki: fail, q = " + str(q))
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elif result == -1:
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elif result == -1:
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print "Przytycki: not applicable, q = " + str(q)
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print("Przytycki: not applicable, q = " + str(q))
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else:
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else:
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print "Przytycki: pass, q = " + str(q)
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print("Przytycki: pass, q = " + str(q))
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class PrzytyckiTester(object):
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class PrzytyckiTester(object):
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@ -452,19 +451,20 @@ def check_criteria(name, pd_code, f_homfly_in=None):
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return tester
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return tester
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def get_naik_number(p, q):
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def get_naik_number(q, p):
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'''
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'''
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Calculate the smallest integer i such that p^i == +/-1 mod q.
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Calculate the smallest integer i = [q, p] 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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Signum of i shows whether rest is -1 or 1
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'''
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'''
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if gcd(q, p) > 1:
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if gcd(q, p) > 1:
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return 0
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return 0
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p_power = p
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p_power = p
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for i in xrange(1, sys.maxint):
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for i in range(1, sys.maxsize):
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pq = p_power % q
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qp = p_power % q
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if pq == 1:
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if qp == 1:
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return i
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return i
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if pq == q - 1:
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if qp == q - 1:
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return -i
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return -i
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p_power *= p
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p_power *= p
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@ -500,8 +500,7 @@ def check_11_to_15(f_out, f_homfly_out=None, f_homfly_in=None):
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pd_code = parse_pd_code(f.readline())
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pd_code = parse_pd_code(f.readline())
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line = f.readline()
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line = f.readline()
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tester = check_criteria(name, pd_code, f_homfly_in)
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tester = check_criteria(name, pd_code, f_homfly_in)
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if tester is None:
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if tester is not None:
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continue
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tester.save_results(f_out)
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tester.save_results(f_out)
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@ -514,17 +513,24 @@ def check_up_to_10(f_out, f_homfly_in=None):
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pd_code = parse_pd_code(str(line[1]))
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pd_code = parse_pd_code(str(line[1]))
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line = f.readline()
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line = f.readline()
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tester = check_criteria(name, pd_code, f_homfly_in)
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tester = check_criteria(name, pd_code, f_homfly_in)
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if tester is None:
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if tester is not None:
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continue
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tester.save_results(f_out)
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tester.save_results(f_out)
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def test_all(f_out, f_homfly_in=None):
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def test_all(f_out, f_homfly_in=None):
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check_up_to_10(f_out, f_homfly_in)
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check_up_to_10(f_out, f_homfly_in)
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if f_out is not None:
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f_out.flush()
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f_out.flush()
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check_11_to_15(f_out, f_homfly_in)
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check_11_to_15(f_out, f_homfly_in)
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def main():
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if settings.input_file_with_homflypt:
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with open(settings.f_results_out, 'w') as f_out,\
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open(settings.f_homfly_lm_in, 'r') as f_homfly_in:
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test_all(f_out, f_homfly_in)
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else:
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with open(settings.f_results_out, 'w') as f_out:
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test_all(f_out)
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if __name__ == '__main__':
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if __name__ == '__main__':
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@ -533,10 +539,10 @@ if __name__ == '__main__':
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R.<t> = LaurentPolynomialRing(ZZ)
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R.<t> = LaurentPolynomialRing(ZZ)
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prime_numbers = Primes()
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prime_numbers = Primes()
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naik_number_dict = {}
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naik_number_dict = {}
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if settings.input_file_with_homflypt:
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if '__file__' in globals():
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with open(settings.f_results_out, 'w') as f_out,\
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main()
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open(settings.f_homfly_lm_in, 'r') as f_homfly_in:
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else:
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test_all(f_out, f_homfly_in)
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S.<a, z> = LaurentPolynomialRing(ZZ)
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else:
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R.<t> = LaurentPolynomialRing(ZZ)
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with open(settings.f_results_out, 'w') as f_out:
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prime_numbers = Primes()
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test_all(f_out)
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naik_number_dict = {}
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@ -1,3 +1,5 @@
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|||||||
|
#!/usr/bin/python
|
||||||
|
|
||||||
# Copyright (c) 2018: Maria Marchwicka, Wojciech Politarczyk.
|
# Copyright (c) 2018: Maria Marchwicka, Wojciech Politarczyk.
|
||||||
# This program is free software: you can redistribute it and/or modify
|
# This program is free software: you can redistribute it and/or modify
|
||||||
# it under the terms of the GNU General Public License as published by
|
# it under the terms of the GNU General Public License as published by
|
||||||
@ -19,6 +21,15 @@ import numpy as np
|
|||||||
import warnings
|
import warnings
|
||||||
|
|
||||||
|
|
||||||
|
if not os.path.isfile('knots_periodicity.py'):
|
||||||
|
os.system('sage --preparse periodicity.sage')
|
||||||
|
os.system('mv periodicity.sage.py knots_periodicity.py')
|
||||||
|
from knots_periodicity import get_t_polynomial, \
|
||||||
|
parse_pd_code, parse_knot_name
|
||||||
|
import knots_periodicity
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
class MySettings(object):
|
class MySettings(object):
|
||||||
|
|
||||||
def __init__(self):
|
def __init__(self):
|
||||||
@ -32,14 +43,14 @@ class MySettings(object):
|
|||||||
self.f_results_out = os.path.join(os.getcwd(), "results.out")
|
self.f_results_out = os.path.join(os.getcwd(), "results.out")
|
||||||
self.f_old_results = os.path.join(os.getcwd(), "old_results.input")
|
self.f_old_results = os.path.join(os.getcwd(), "old_results.input")
|
||||||
|
|
||||||
self.periods = [9]
|
# self.periods = [9]
|
||||||
|
|
||||||
# self.periods = [3, 5, 7, 9, 11]
|
self.periods = [3, 5, 7, 9, 11]
|
||||||
self.set_to_check = self.get_set()
|
self.set_to_check = self.get_set()
|
||||||
|
|
||||||
# check only knots from defined set
|
# check only knots from defined set
|
||||||
self.only_chosen = True
|
self.only_chosen = True
|
||||||
self.only_chosen = False
|
# self.only_chosen = False
|
||||||
|
|
||||||
self.debugging = True
|
self.debugging = True
|
||||||
# self.debugging = False
|
# self.debugging = False
|
||||||
@ -56,8 +67,8 @@ class MySettings(object):
|
|||||||
self.only_periods = True
|
self.only_periods = True
|
||||||
self.only_periods = False
|
self.only_periods = False
|
||||||
|
|
||||||
self.print_results = False
|
|
||||||
self.print_results = True
|
self.print_results = True
|
||||||
|
# self.print_results = False
|
||||||
|
|
||||||
# saving HOMFLYPT polynomials into self.f_homfly_lm_out
|
# saving HOMFLYPT polynomials into self.f_homfly_lm_out
|
||||||
self.save_homfly = True
|
self.save_homfly = True
|
||||||
@ -68,7 +79,7 @@ class MySettings(object):
|
|||||||
# self.input_file_with_homflypt = False
|
# self.input_file_with_homflypt = False
|
||||||
|
|
||||||
self.check_old_results = True
|
self.check_old_results = True
|
||||||
# self.check_old_results = False
|
self.check_old_results = False
|
||||||
|
|
||||||
if self.input_file_with_homflypt:
|
if self.input_file_with_homflypt:
|
||||||
if not os.path.isfile(self.f_homfly_lm_in):
|
if not os.path.isfile(self.f_homfly_lm_in):
|
||||||
@ -598,309 +609,10 @@ class MySettings(object):
|
|||||||
|
|
||||||
set_to_check |= set(["10_123"])
|
set_to_check |= set(["10_123"])
|
||||||
set_to_check |= set(["15n166130"])
|
set_to_check |= set(["15n166130"])
|
||||||
set_to_check = set(self.fails_dict.keys())
|
|
||||||
|
|
||||||
return set_to_check
|
return set_to_check
|
||||||
|
|
||||||
|
class PeriodicityTester(knots_periodicity.PeriodicityTester):
|
||||||
class PeriodicityTester(object):
|
|
||||||
|
|
||||||
def __init__(self, name, pd_code, A=None, f_homfly_in=None):
|
|
||||||
|
|
||||||
self.results = []
|
|
||||||
'''
|
|
||||||
To results for each period q a list in following form will be appended:
|
|
||||||
[q, murasugi, naik_1, naik_2, borodzik, przytycki]
|
|
||||||
Crierion is set to be:
|
|
||||||
-1 if it is not applicable (details in check_naik_2, check_przytycki,
|
|
||||||
1 if criterion doesn't exclude periodic,
|
|
||||||
0 if criterion excludes periodicity.
|
|
||||||
murasugi, naik_1, naik_2 or borodzik is also set to be:
|
|
||||||
2 if alexander_polynomial == 1.
|
|
||||||
0 if previous criterion in the list is 0.
|
|
||||||
'''
|
|
||||||
|
|
||||||
self.name = name
|
|
||||||
self.pd_code = pd_code
|
|
||||||
|
|
||||||
self.smith = None
|
|
||||||
self.reset_results()
|
|
||||||
|
|
||||||
if pd_code is not None:
|
|
||||||
self.K = Link(pd_code)
|
|
||||||
self.seifert = self.K.seifert_matrix()
|
|
||||||
else:
|
|
||||||
self.seifert = A
|
|
||||||
# delta := Alexander polynomial
|
|
||||||
delta = (self.seifert.transpose() - t * self.seifert).determinant()
|
|
||||||
self.delta = delta.shift(-delta.exponents()[0])
|
|
||||||
self.delta_factors = self.set_delta_factors()
|
|
||||||
self.przytycki_tester = self.get_przytycki_tester(f_homfly_in)
|
|
||||||
|
|
||||||
def reset_results(self):
|
|
||||||
self.murasugi = 0
|
|
||||||
self.naik_1 = 0
|
|
||||||
self.naik_2 = 0
|
|
||||||
self.borodzik = 0
|
|
||||||
self.przytycki = 0
|
|
||||||
self.murasugi_fulfilling = set()
|
|
||||||
self.naik_1_fulfilling = []
|
|
||||||
self.naik_2_fulfilling = []
|
|
||||||
|
|
||||||
def set_smith(self):
|
|
||||||
symetric_from_seifert = self.seifert + self.seifert.transpose()
|
|
||||||
assert symetric_from_seifert.determinant() != 0, \
|
|
||||||
"The determinant of A + A^T is zero."
|
|
||||||
self.smith = symetric_from_seifert.smith_form()
|
|
||||||
D, U, V = self.smith
|
|
||||||
self.diagonal = D.diagonal()
|
|
||||||
self.maximum_in_diagonal = max(self.diagonal)
|
|
||||||
C = U.inverse()
|
|
||||||
E_inverse = V
|
|
||||||
self.C_tran_E_inv_D_inv = C.transpose() * E_inverse * D.inverse()
|
|
||||||
self.matrix_C = C
|
|
||||||
self.matrix_E_inverse = E_inverse
|
|
||||||
|
|
||||||
def get_przytycki_tester(self, f_homfly_in):
|
|
||||||
if self.pd_code is not None:
|
|
||||||
try:
|
|
||||||
return PrzytyckiTester(self.K, self.name, f_homfly_in)
|
|
||||||
except ImportError as e:
|
|
||||||
if settings.debugging:
|
|
||||||
print "Error by checking Przytycki criterion.\n" + str(e)
|
|
||||||
return None
|
|
||||||
|
|
||||||
def get_C_tran_E_inv_D_inv(self):
|
|
||||||
if self.smith is None:
|
|
||||||
self.set_smith()
|
|
||||||
return self.C_tran_E_inv_D_inv
|
|
||||||
|
|
||||||
def get_maximum_in_diagonal(self):
|
|
||||||
if self.smith is None:
|
|
||||||
self.set_smith()
|
|
||||||
return self.maximum_in_diagonal
|
|
||||||
|
|
||||||
def set_delta_factors(self):
|
|
||||||
# find all delta (alexander polynomial) factors
|
|
||||||
lst_of_factors = [[f[0]] * f[1] for f in self.delta.factor()]
|
|
||||||
# flattening a list
|
|
||||||
lst_of_factors = [el for sublist in lst_of_factors for el in sublist]
|
|
||||||
delta_candidates = set()
|
|
||||||
for s in get_subsets(lst_of_factors):
|
|
||||||
d = t^0
|
|
||||||
for el in s:
|
|
||||||
d *= el
|
|
||||||
delta_candidates.add(d)
|
|
||||||
return delta_candidates
|
|
||||||
|
|
||||||
def check_criteria_for_period(self, q):
|
|
||||||
|
|
||||||
self.reset_results()
|
|
||||||
self.przytycki = self.check_przytycki(q)
|
|
||||||
|
|
||||||
if self.delta == 1:
|
|
||||||
self.murasugi = 2
|
|
||||||
self.naik_1 = 2
|
|
||||||
self.naik_2 = 2
|
|
||||||
self.borodzik = 2
|
|
||||||
return 2
|
|
||||||
|
|
||||||
self.murasugi = self.check_murasugi(q)
|
|
||||||
self.naik_1 = self.check_naik_1(q)
|
|
||||||
self.naik_2 = self.check_naik_2(q)
|
|
||||||
self.borodzik = self.check_borodzik(q)
|
|
||||||
|
|
||||||
if settings.debugging:
|
|
||||||
print ("\n" + "#" * 30 + " Calculations for knot " + self.name +
|
|
||||||
" and q = " + str(q) + " " + "#" * 30 + "\n")
|
|
||||||
self.print_data_for_murasugi(q)
|
|
||||||
self.print_data_for_naik_1(q)
|
|
||||||
self.print_data_for_naik_2(q)
|
|
||||||
self.print_data_for_borodzik(q)
|
|
||||||
|
|
||||||
return self.borodzik * self.przytycki
|
|
||||||
|
|
||||||
def check_murasugi(self, q):
|
|
||||||
'''
|
|
||||||
Select these delta factors and natural number r such that:
|
|
||||||
delta = delta_prime^q * (1 + t^1 + ... + t^(r-1))^(q-1) mod q_factor
|
|
||||||
where "delta_prime" is a delta factor.
|
|
||||||
'''
|
|
||||||
q_factor = factor(q)[0][0]
|
|
||||||
quotient_delta = self.delta.change_ring(GF(q_factor))
|
|
||||||
# Underlying polynomial of quotient_delta:
|
|
||||||
quotient_delta = quotient_delta.polynomial_construction()[0]
|
|
||||||
delta_degree = quotient_delta.degree()
|
|
||||||
|
|
||||||
for candidate in self.delta_factors:
|
|
||||||
quotient_candidate = candidate.change_ring(GF(q_factor))
|
|
||||||
power_candidate = quotient_candidate^q
|
|
||||||
power_candidate = power_candidate.polynomial_construction()[0]
|
|
||||||
# (r - 1) - possible t-polynomial degree
|
|
||||||
r = (delta_degree - power_candidate.degree()) / (q - 1) + 1
|
|
||||||
if r < 1 or not r.is_integer():
|
|
||||||
continue
|
|
||||||
t_polynomial = get_t_polynomial(q, r)
|
|
||||||
right_side = t_polynomial * power_candidate
|
|
||||||
if quotient_delta != right_side and -quotient_delta != right_side:
|
|
||||||
continue
|
|
||||||
self.murasugi_fulfilling.add((candidate, r))
|
|
||||||
|
|
||||||
return int(bool(self.murasugi_fulfilling))
|
|
||||||
|
|
||||||
def check_naik_1_candidate(self, delta_prime, delta_evaluated, q):
|
|
||||||
|
|
||||||
t_delta = delta_evaluated / delta_prime(-1)
|
|
||||||
t_delta_dict = {f[0]: f[1] for f in factor(t_delta)}
|
|
||||||
t_delta_factors = [f for f in t_delta_dict.keys()
|
|
||||||
if f != 2 and gcd(q, f) == 1]
|
|
||||||
for f in t_delta_factors:
|
|
||||||
q_f = naik_number_dict.setdefault((q, f), get_naik_number(q, f))
|
|
||||||
if not (t_delta_dict[f] / (2 * q_f)).is_integer():
|
|
||||||
return None
|
|
||||||
return t_delta_factors
|
|
||||||
|
|
||||||
def check_naik_1(self, q):
|
|
||||||
'''
|
|
||||||
For each delta' find a set P of prime numbers p such that:
|
|
||||||
gcd(q, p) == 1, p != 2 and p| t_delta, t_delta = delta(-1)/delta'(-1).
|
|
||||||
Check if all p factors of t_delta has multiplicity divisible by 2*[q|p].
|
|
||||||
If it holds for at least one delta' candidate, set naik_1 = True.
|
|
||||||
'''
|
|
||||||
# Proposition 2.7.
|
|
||||||
delta_evaluated = self.delta(-1)
|
|
||||||
|
|
||||||
for delta_prime, _ in self.murasugi_fulfilling:
|
|
||||||
t_delta_factors = self.check_naik_1_candidate(delta_prime,
|
|
||||||
delta_evaluated, q)
|
|
||||||
if t_delta_factors is not None:
|
|
||||||
self.naik_1_fulfilling.append((delta_prime, t_delta_factors))
|
|
||||||
|
|
||||||
return int(bool(self.naik_1_fulfilling))
|
|
||||||
|
|
||||||
def check_naik_2_candidate(self, q, p_list):
|
|
||||||
delta_prime_bases = []
|
|
||||||
maximum_in_diagonal = self.get_maximum_in_diagonal()
|
|
||||||
for p in p_list:
|
|
||||||
q_p = naik_number_dict[(q, p)]
|
|
||||||
bases_for_p_torsion = []
|
|
||||||
factor_power = p
|
|
||||||
# find all p^k torsion parts
|
|
||||||
while (maximum_in_diagonal / factor_power).is_integer():
|
|
||||||
basis_for_p_k_part = []
|
|
||||||
for el in self.diagonal:
|
|
||||||
to_be_append = el / factor_power
|
|
||||||
is_int = (to_be_append / p).is_integer()
|
|
||||||
if to_be_append.is_integer() and not is_int:
|
|
||||||
basis_for_p_k_part.append(to_be_append)
|
|
||||||
else:
|
|
||||||
basis_for_p_k_part.append(0)
|
|
||||||
len_non_zero = sum(x != 0 for x in basis_for_p_k_part)
|
|
||||||
# check if dimension is multiple of 2 * naik_number
|
|
||||||
if not (len_non_zero / (2 * q_p)).is_integer():
|
|
||||||
return None
|
|
||||||
factor_power *= p
|
|
||||||
bases_for_p_torsion.append(basis_for_p_k_part)
|
|
||||||
delta_prime_bases.append((p, bases_for_p_torsion))
|
|
||||||
return delta_prime_bases
|
|
||||||
|
|
||||||
def check_naik_2(self, q):
|
|
||||||
'''
|
|
||||||
For each delta' consider a set P of primes p such that: gcd(p, q) == 1,
|
|
||||||
p != 2, p| delta(-1)/delta'(-1) (self.naik_1_fulfilling) and p is not
|
|
||||||
a factor of delta'(-1). Check if dimension of p^k torsion part
|
|
||||||
is divisible by 2*[p|q] for all k and all p from P.
|
|
||||||
If it holds for at least one delta' candidate, we set naik_2 to be True.
|
|
||||||
In particular naik_2 is set to be -1 if the criterion passes,
|
|
||||||
but only in cases where P is an empty set.
|
|
||||||
'''
|
|
||||||
# Proposition 2.8.
|
|
||||||
for delta_prime, p_list in self.naik_1_fulfilling:
|
|
||||||
delta_prime_factors = set([d[0] for d in factor(delta_prime(-1))])
|
|
||||||
p_list = [p for p in p_list if p not in delta_prime_factors]
|
|
||||||
|
|
||||||
if not p_list:
|
|
||||||
self.naik_2 = -1
|
|
||||||
self.borodzik = -1
|
|
||||||
continue
|
|
||||||
|
|
||||||
delta_prime_bases = self.check_naik_2_candidate(q, p_list)
|
|
||||||
if delta_prime_bases is not None:
|
|
||||||
self.naik_2_fulfilling.append((delta_prime,
|
|
||||||
delta_prime_bases))
|
|
||||||
if self.naik_2_fulfilling:
|
|
||||||
return 1
|
|
||||||
return self.naik_2
|
|
||||||
|
|
||||||
def check_borodzik(self, q):
|
|
||||||
'''
|
|
||||||
Consider all delta' that meet criterion Naik 2.
|
|
||||||
For all p from a set P (defined as in check_naik_2)
|
|
||||||
and all k consider p^k torsion part.
|
|
||||||
For each p^k torsion check if eta == epsilon_1 * epsilon_2
|
|
||||||
(see check_borodzik_candidate()).
|
|
||||||
If it holds for at least one delta' candidate, set borodzik to be True.
|
|
||||||
In particular borodzik is set to be -1 if the criterion passes,
|
|
||||||
but only in cases where P is an empty set.
|
|
||||||
'''
|
|
||||||
|
|
||||||
for delta_prime, delta_prime_bases in self.naik_2_fulfilling:
|
|
||||||
borodzik_pass = True
|
|
||||||
for p, bases_for_p in delta_prime_bases:
|
|
||||||
# if len(bases_for_p) > 1:
|
|
||||||
# print "HURA" # more than one p^k part - not found yet
|
|
||||||
if not self.check_borodzik_candidate(q, p, bases_for_p):
|
|
||||||
borodzik_pass = False
|
|
||||||
break
|
|
||||||
if borodzik_pass:
|
|
||||||
return 1
|
|
||||||
return self.borodzik
|
|
||||||
|
|
||||||
def check_borodzik_candidate(self, q, p, bases):
|
|
||||||
'''
|
|
||||||
For each p^k torsion check if eta == epsilon_1 * epsilon_2.
|
|
||||||
If determinant of corsesponding matrix P is square modulo p, then:
|
|
||||||
episilon_1 = 1, else: episilon_1 = -1.
|
|
||||||
If p == 3 mod(4) and a rank of p^k torsion part n == 2 mod(4), then:
|
|
||||||
epsilon_2 = -1, else: epsilon_2 = 1.
|
|
||||||
eta = naik_sign^d, where d = n / (2 * [p, q]).
|
|
||||||
If p^([p, q]) % q == 1, then: naik_sign = 1, else: naik_sign = -1.
|
|
||||||
'''
|
|
||||||
for k, p_k_basis in enumerate(bases):
|
|
||||||
X = np.diagflat(p_k_basis)
|
|
||||||
# columns that up to zero (element in diagonal is zero):
|
|
||||||
zero_columns = np.nonzero(X.sum(axis=0) == 0)
|
|
||||||
X = np.delete(X, zero_columns, axis=1)
|
|
||||||
n = X.shape[1]
|
|
||||||
X = matrix(X)
|
|
||||||
P = p^(k + 1) * X.transpose() * self.get_C_tran_E_inv_D_inv() * X
|
|
||||||
P_det = P.determinant()
|
|
||||||
if P_det % p == 0:
|
|
||||||
raise ValueError("P determinant is 0 modulo p.")
|
|
||||||
|
|
||||||
if p % 4 == 3 and n % 4 == 2: # epsilon_1
|
|
||||||
epsilon = -1
|
|
||||||
else:
|
|
||||||
epsilon = 1
|
|
||||||
|
|
||||||
if not mod(P_det, p).is_square():
|
|
||||||
epsilon *= -1 # epsilon = epsilon_1 * epsilon_2
|
|
||||||
|
|
||||||
q_p = naik_number_dict[(q, p)]
|
|
||||||
d = n / (2 * q_p)
|
|
||||||
# sign(q_p) - whether rest is -1 or 1
|
|
||||||
if sign(q_p)^d != epsilon:
|
|
||||||
return False
|
|
||||||
return True
|
|
||||||
|
|
||||||
def check_przytycki(self, q):
|
|
||||||
if self.przytycki_tester is not None and q in prime_numbers:
|
|
||||||
try:
|
|
||||||
return self.przytycki_tester.check_congruence(q)
|
|
||||||
except (AttributeError, OverflowError) as e:
|
|
||||||
pass
|
|
||||||
return -1
|
|
||||||
|
|
||||||
def save_results(self, f_out, f_homfly_out=None):
|
def save_results(self, f_out, f_homfly_out=None):
|
||||||
for result in self.results:
|
for result in self.results:
|
||||||
line_to_write = self.name + "," + ",".join(map(str, result))
|
line_to_write = self.name + "," + ",".join(map(str, result))
|
||||||
@ -913,15 +625,15 @@ class PeriodicityTester(object):
|
|||||||
old_results = [int(x) for x in line[2:]]
|
old_results = [int(x) for x in line[2:]]
|
||||||
# if old_results[:-1] != result[1:-1]:
|
# if old_results[:-1] != result[1:-1]:
|
||||||
if old_results[:] != result[1:]:
|
if old_results[:] != result[1:]:
|
||||||
print ("#" * 30 + " ERROR " + line[0] + " " +
|
print("#" * 30 + " ERROR " + line[0] + " " +
|
||||||
"#" * 30)
|
"#" * 30)
|
||||||
print "q = " + line[1]
|
print("q = " + line[1])
|
||||||
print "result " + str(result[1:])
|
print("result " + str(result[1:]))
|
||||||
print "old_results " + str(old_results)
|
print("old_results " + str(old_results))
|
||||||
break
|
break
|
||||||
line = f_old_results.readline()
|
line = f_old_results.readline()
|
||||||
if not line:
|
if not line:
|
||||||
print "No data to compare."
|
print("No data to compare.")
|
||||||
f_out.writelines(line_to_write + "\n")
|
f_out.writelines(line_to_write + "\n")
|
||||||
|
|
||||||
if self.przytycki_tester is not None and f_homfly_out is not None:
|
if self.przytycki_tester is not None and f_homfly_out is not None:
|
||||||
@ -930,76 +642,36 @@ class PeriodicityTester(object):
|
|||||||
f_homfly_out.writelines(line_to_write)
|
f_homfly_out.writelines(line_to_write)
|
||||||
|
|
||||||
def print_results(self):
|
def print_results(self):
|
||||||
|
super().print_results()
|
||||||
print "\n" + "#" * 15 + " " + str(self.name) + " " + "#" * 15
|
|
||||||
if self.name in settings.periods_dict:
|
if self.name in settings.periods_dict:
|
||||||
print "periods: " + str(settings.periods_dict[self.name])
|
print("periods: " + str(settings.periods_dict[self.name]))
|
||||||
|
|
||||||
for result in self.results:
|
|
||||||
|
|
||||||
q = result[0]
|
|
||||||
print
|
|
||||||
self.print_przytycki_result(q, result[5])
|
|
||||||
|
|
||||||
if result[1] == 2:
|
|
||||||
print "Alexander polynomial is 1"
|
|
||||||
continue
|
|
||||||
|
|
||||||
if not result[1]:
|
|
||||||
print "\t\tMurasugi: fail, q = " + str(q)
|
|
||||||
continue
|
|
||||||
|
|
||||||
print "Murasugi: pass, q = " + str(q)
|
|
||||||
|
|
||||||
if not result[2]:
|
|
||||||
print "\t\tNaik 1: fail, q = " + str(q)
|
|
||||||
continue
|
|
||||||
|
|
||||||
print "Naik 1: pass, q = " + str(q)
|
|
||||||
|
|
||||||
if not result[3]:
|
|
||||||
print "\t\tNaik 2: fail, q = " + str(q)
|
|
||||||
continue
|
|
||||||
|
|
||||||
if result[3] == -1:
|
|
||||||
print "Naik 2: not applicable, q = " + str(q)
|
|
||||||
continue
|
|
||||||
|
|
||||||
print "Naik 2: pass, q = " + str(q)
|
|
||||||
|
|
||||||
if not result[4]:
|
|
||||||
print ("\t\tBorodzik: fail, q = " + str(q))
|
|
||||||
continue
|
|
||||||
|
|
||||||
if result[4] == -1:
|
|
||||||
print ("Borodzik: not applicable, q = " + str(q))
|
|
||||||
continue
|
|
||||||
|
|
||||||
print ("Borodzik: pass, q = " + str(q))
|
|
||||||
|
|
||||||
def print_przytycki_result(self, q, result):
|
def print_przytycki_result(self, q, result):
|
||||||
if not result:
|
if not result:
|
||||||
print "\t\tPrzytycki: fail, q = " + str(q)
|
print("\t\tPrzytycki: fail, q = " + str(q))
|
||||||
elif result == -1:
|
elif result == -1:
|
||||||
print "Przytycki: not applicable, q = " + str(q)
|
print("Przytycki: not applicable, q = " + str(q))
|
||||||
else:
|
else:
|
||||||
print "Przytycki: pass, q = " + str(q)
|
print("Przytycki: pass, q = " + str(q))
|
||||||
|
|
||||||
def print_data_for_murasugi(self, q):
|
def print_data_for_murasugi(self, q):
|
||||||
|
|
||||||
if self.murasugi:
|
if self.murasugi:
|
||||||
print ("\n" + "#" * 30 + " Knot " + str(self.name) +
|
print("\n" + "#" * 30 + " Knot " + str(self.name) +
|
||||||
" passes Murasugi condition for q = " +
|
" passes Murasugi condition for q = " +
|
||||||
str(q) + " " + "#" * 30)
|
str(q) + " " + "#" * 30)
|
||||||
else:
|
else:
|
||||||
print ("\nKnot " + str(self.name) +
|
print("\nKnot " + str(self.name) +
|
||||||
" fails Murasugi condition for q = " + str(q))
|
" fails Murasugi condition for q = " + str(q))
|
||||||
|
|
||||||
quotient_delta = self.delta.change_ring(GF(q))
|
quotient_delta = self.delta.change_ring(GF(q))
|
||||||
quotient_delta = quotient_delta.polynomial_construction()[0]
|
quotient_delta = quotient_delta.polynomial_construction()[0]
|
||||||
print "delta: " + str(self.delta)
|
print("delta: \t" + str(self.delta))
|
||||||
print "delta factors: " + str(self.delta.factor())
|
if self.delta == 1:
|
||||||
print "delta mod q = " + str(quotient_delta)
|
print("delta factors: " + str([1]))
|
||||||
|
else:
|
||||||
|
print("delta factors: " + str(self.delta.factor()))
|
||||||
|
print("delta mod q = " + str(quotient_delta))
|
||||||
delta_degree = quotient_delta.degree()
|
delta_degree = quotient_delta.degree()
|
||||||
self.print_murasugi_fulfilling(q)
|
self.print_murasugi_fulfilling(q)
|
||||||
# self.print_candidates_that_fail_murasugi(q)
|
# self.print_candidates_that_fail_murasugi(q)
|
||||||
@ -1008,17 +680,17 @@ class PeriodicityTester(object):
|
|||||||
quotient_delta = self.delta.change_ring(GF(q))
|
quotient_delta = self.delta.change_ring(GF(q))
|
||||||
quotient_delta = quotient_delta.polynomial_construction()[0]
|
quotient_delta = quotient_delta.polynomial_construction()[0]
|
||||||
delta_degree = quotient_delta.degree()
|
delta_degree = quotient_delta.degree()
|
||||||
print ("\nNumber of candidates that pass Murasugi = " +
|
print("\nNumber of candidates that pass Murasugi = " +
|
||||||
str(len(self.murasugi_fulfilling)))
|
str(len(self.murasugi_fulfilling)))
|
||||||
for i, (delta_prime, r) in enumerate(self.murasugi_fulfilling):
|
for i, (delta_prime, r) in enumerate(self.murasugi_fulfilling):
|
||||||
print "\n" + str(i + 1) + ". delta_prime:\t" + str(delta_prime)
|
print("\n" + str(i + 1) + ". delta_prime:\t" + str(delta_prime))
|
||||||
t_polynomial = get_t_polynomial(q, r)
|
t_polynomial = get_t_polynomial(q, r)
|
||||||
print "polynomial^(q-1) = " + str(t_polynomial)
|
print("polynomial^(q-1) = " + str(t_polynomial))
|
||||||
right_side = t_polynomial * delta_prime^q
|
right_side = t_polynomial * delta_prime^q
|
||||||
print "*" * 50
|
print("*" * 50)
|
||||||
print "delta == delta_prime^q * polynomial^(q-1) mod q"
|
print("delta == delta_prime^q * polynomial^(q-1) mod q")
|
||||||
print "right side:\t" + str(right_side.factor())
|
# print("right side:\t" + str(right_side.factor()))
|
||||||
print "left side:\t" + str(quotient_delta.factor())
|
# print("left side: \t" + str(quotient_delta.factor()))
|
||||||
|
|
||||||
def print_candidates_that_fail_murasugi(self, q):
|
def print_candidates_that_fail_murasugi(self, q):
|
||||||
quotient_delta = self.delta.change_ring(GF(q))
|
quotient_delta = self.delta.change_ring(GF(q))
|
||||||
@ -1035,65 +707,65 @@ class PeriodicityTester(object):
|
|||||||
if (quotient_delta == right_side or
|
if (quotient_delta == right_side or
|
||||||
(-quotient_delta) == right_side):
|
(-quotient_delta) == right_side):
|
||||||
continue
|
continue
|
||||||
print "\nFor candidate = " + str(candidate)
|
print("\nFor candidate = " + str(candidate))
|
||||||
print "quotient_candidate = " + str(quotient_candidate)
|
print( "quotient_candidate = " + str(quotient_candidate))
|
||||||
print "candidate^q = " + str(power_candidate)
|
print("candidate^q = " + str(power_candidate))
|
||||||
print "shifted = " + str(shifted_candidate)
|
print("shifted = " + str(shifted_candidate))
|
||||||
print "delta degree = " + str(delta_degree)
|
print("delta degree = " + str(delta_degree))
|
||||||
print "candidate^q degree " + str(shifted_candidate.degree())
|
print("candidate^q degree " + str(shifted_candidate.degree()))
|
||||||
print "r = " + str(r)
|
print("r = " + str(r))
|
||||||
if r > 0 and r.is_integer():
|
if r > 0 and r.is_integer():
|
||||||
print "right_side = " + str(right_side)
|
print("right_side = " + str(right_side))
|
||||||
print "delta mod q = " + str(quotient_delta)
|
print("delta mod q = " + str(quotient_delta))
|
||||||
|
|
||||||
def print_data_for_naik_1(self, q):
|
def print_data_for_naik_1(self, q):
|
||||||
if not self.murasugi:
|
if not self.murasugi:
|
||||||
return None
|
return None
|
||||||
if not self.naik_1:
|
if not self.naik_1:
|
||||||
print ("\nKnot " + str(self.name) +
|
print("\nKnot " + str(self.name) +
|
||||||
" fails Naik 1 condition for q = " + str(q))
|
" fails Naik 1 condition for q = " + str(q))
|
||||||
else:
|
else:
|
||||||
print ("\n" + "#" * 30 + " Knot " + str(self.name) +
|
print("\n" + "#" * 30 + " Knot " + str(self.name) +
|
||||||
" passes Naik 1 condition for q = " + str(q) +
|
" passes Naik 1 condition for q = " + str(q) +
|
||||||
" " + "#" * 30)
|
" " + "#" * 30)
|
||||||
print "delta: " + str(self.delta)
|
print("delta: " + str(self.delta))
|
||||||
print "delta at -1: " + str(self.delta(-1))
|
print("delta at -1: " + str(self.delta(-1)))
|
||||||
print "factors for evaluated: " + str(self.delta(-1).factor())
|
print("factors for evaluated: " + str(self.delta(-1).factor()))
|
||||||
self.print_naik_1_fulfilling(q)
|
self.print_naik_1_fulfilling(q)
|
||||||
|
|
||||||
def print_naik_1_fulfilling(self, q):
|
def print_naik_1_fulfilling(self, q):
|
||||||
print ("\nNumber of candidates that pass Naik 1 = " +
|
print("\nNumber of candidates that pass Naik 1 = " +
|
||||||
str(len(self.naik_1_fulfilling)))
|
str(len(self.naik_1_fulfilling)))
|
||||||
for delta_prime, p_list in self.naik_1_fulfilling:
|
for delta_prime, p_list in self.naik_1_fulfilling:
|
||||||
print "delta prime: " + str(delta_prime)
|
print("delta prime: " + str(delta_prime))
|
||||||
print "delta prime at -1: " + str(delta_prime(-1))
|
print("delta prime at -1: " + str(delta_prime(-1)))
|
||||||
t_delta = self.delta(-1)/delta_prime(-1)
|
t_delta = self.delta(-1)/delta_prime(-1)
|
||||||
print "delta/delta_prime(-1):\t\t" + str(t_delta)
|
print("delta/delta_prime(-1):\t\t" + str(t_delta))
|
||||||
print "delta/delta_prime(-1) factors:\t" + str(t_delta.factor())
|
print("delta/delta_prime(-1) factors:\t" + str(t_delta.factor()))
|
||||||
if not p_list:
|
if not p_list:
|
||||||
print "List of factors was empty."
|
print("List of factors was empty.")
|
||||||
for p in p_list:
|
for p in p_list:
|
||||||
g = abs(naik_number_dict[(q, p)])
|
g = abs(naik_number_dict[(q, p)])
|
||||||
print "factor of del/del'(-1): " + str(p)
|
print("factor of del/del'(-1): " + str(p))
|
||||||
print "Naik number: " + str(g)
|
print("Naik number: " + str(g))
|
||||||
print "2 * Naik number:\t" + str(2 * g)
|
print("2 * Naik number:\t" + str(2 * g))
|
||||||
test_naik_number = p^g % q
|
test_naik_number = p^g % q
|
||||||
print (str(p) + "^" + str(g) + " % " + str(q) + " = " +
|
print(str(p) + "^" + str(g) + " % " + str(q) + " = " +
|
||||||
str(test_naik_number) + " = " +
|
str(test_naik_number) + " = " +
|
||||||
str(test_naik_number - q))
|
str(test_naik_number - q))
|
||||||
t_delta_dict = {i[0]: i[1] for i in factor(t_delta)}
|
t_delta_dict = {i[0]: i[1] for i in factor(t_delta)}
|
||||||
print "The power of factor:\t" + str(t_delta_dict[p])
|
print("The power of factor:\t" + str(t_delta_dict[p]))
|
||||||
|
|
||||||
def print_data_for_naik_2(self, q):
|
def print_data_for_naik_2(self, q):
|
||||||
if not self.naik_1:
|
if not self.naik_1:
|
||||||
return None
|
return None
|
||||||
if not self.naik_2:
|
if not self.naik_2:
|
||||||
return None
|
return None
|
||||||
print ("\n" + "#" * 30 + " Knot " + str(self.name) +
|
print("\n" + "#" * 30 + " Knot " + str(self.name) +
|
||||||
" passes Naik 2 condition for q = " + str(q) + " " + "#" * 30)
|
" passes Naik 2 condition for q = " + str(q) + " " + "#" * 30)
|
||||||
print "delta:\t\t\t" + str(self.delta)
|
print("delta:\t\t\t" + str(self.delta))
|
||||||
print "delta at -1:\t\t" + str(self.delta(-1))
|
print("delta at -1:\t\t" + str(self.delta(-1)))
|
||||||
print "factors for evaluated:\t" + str(self.delta(-1).factor())
|
print("factors for evaluated:\t" + str(self.delta(-1).factor()))
|
||||||
if self.naik_2 == -1:
|
if self.naik_2 == -1:
|
||||||
self.print_naik_2_not_applicable(q)
|
self.print_naik_2_not_applicable(q)
|
||||||
return None
|
return None
|
||||||
@ -1104,76 +776,76 @@ class PeriodicityTester(object):
|
|||||||
delta_prime_factors = set([d[0] for d in factor(delta_prime(-1))])
|
delta_prime_factors = set([d[0] for d in factor(delta_prime(-1))])
|
||||||
p_list = [p for p in p_list if p not in delta_prime_factors]
|
p_list = [p for p in p_list if p not in delta_prime_factors]
|
||||||
if not p_list:
|
if not p_list:
|
||||||
print ("\nChecking Naik 2 condition for candidate " +
|
print("\nChecking Naik 2 condition for candidate " +
|
||||||
str(delta_prime) + " and q = " + str(q)) + "."
|
str(delta_prime) + " and q = " + str(q) + ".")
|
||||||
print ("The list of factors was empty or all factors " +
|
print("The list of factors was empty or all factors " +
|
||||||
"were dela'(-1) factors.")
|
"were dela'(-1) factors.")
|
||||||
print "Naik 2 and Borodzik can not exclude periodicity.\n"
|
print("Naik 2 and Borodzik can not exclude periodicity.\n")
|
||||||
|
|
||||||
def print_naik_2_fulfilling(self, q):
|
def print_naik_2_fulfilling(self, q):
|
||||||
for delta_prime, delta_prime_bases in self.naik_2_fulfilling:
|
for delta_prime, delta_prime_bases in self.naik_2_fulfilling:
|
||||||
print "\ndelta prime:\t\t\t" + str(delta_prime)
|
print("\ndelta prime:\t\t\t" + str(delta_prime))
|
||||||
print "delta prime at -1:\t\t" + str(delta_prime(-1))
|
print("delta prime at -1:\t\t" + str(delta_prime(-1)))
|
||||||
t_delta = self.delta(-1)/delta_prime(-1)
|
t_delta = self.delta(-1)/delta_prime(-1)
|
||||||
print "delta/delta_prime(-1):\ " + str(t_delta)
|
print("delta/delta_prime(-1): " + str(t_delta))
|
||||||
print "delta/delta_prime(-1) factors: " + str(t_delta.factor())
|
print("delta/delta_prime(-1) factors: " + str(t_delta.factor()))
|
||||||
|
|
||||||
for p, bases_for_p in delta_prime_bases:
|
for p, bases_for_p in delta_prime_bases:
|
||||||
print "\nfactor p for delta prime:\t\t\t" + str(p)
|
print("\nfactor p for delta prime:\t\t\t" + str(p))
|
||||||
g = abs(naik_number_dict[(q, p)])
|
g = abs(naik_number_dict[(q, p)])
|
||||||
print "Naik number:\t\t" + str(g)
|
print("Naik number:\t\t" + str(g))
|
||||||
print "2 * Naik number:\t" + str(2 * g)
|
print("2 * Naik number:\t" + str(2 * g))
|
||||||
test_naik_number = p^g % q
|
test_naik_number = p^g % q
|
||||||
print (str(p) + "^" + str(g) + " % " + str(q) + " = " +
|
print(str(p) + "^" + str(g) + " % " + str(q) + " = " +
|
||||||
str(test_naik_number) + " = " +
|
str(test_naik_number) + " = " +
|
||||||
str(test_naik_number - q))
|
str(test_naik_number - q))
|
||||||
t_delta_dict = {i[0]: i[1] for i in factor(t_delta)}
|
t_delta_dict = {i[0]: i[1] for i in factor(t_delta)}
|
||||||
print "The power of factor:\t" + str(t_delta_dict[p])
|
print("The power of factor:\t" + str(t_delta_dict[p]))
|
||||||
print "diagonal: " + str(self.diagonal)
|
print("diagonal: " + str(self.diagonal))
|
||||||
print "p^k basis"
|
print("p^k basis")
|
||||||
for k, b in enumerate(bases_for_p):
|
for k, b in enumerate(bases_for_p):
|
||||||
print "k = " + str(k + 1)
|
print("k = " + str(k + 1))
|
||||||
print "basis:\t" + str(b)
|
print("basis:\t" + str(b))
|
||||||
|
|
||||||
def print_data_for_borodzik(self, q):
|
def print_data_for_borodzik(self, q):
|
||||||
|
|
||||||
if self.naik_2 != 1:
|
if self.naik_2 != 1:
|
||||||
return None
|
return None
|
||||||
if self.borodzik:
|
if self.borodzik:
|
||||||
print ("\n" + "#" * 30 + " Knot " + str(self.name) +
|
print("\n" + "#" * 30 + " Knot " + str(self.name) +
|
||||||
" passes Borodzik condition for q = " +
|
" passes Borodzik condition for q = " +
|
||||||
str(q) + " " + "#" * 30)
|
str(q) + " " + "#" * 30)
|
||||||
else:
|
else:
|
||||||
print "%" * 200
|
print("%" * 200)
|
||||||
print ("\nKnot " + str(self.name) +
|
print("\nKnot " + str(self.name) +
|
||||||
" fails Borodzik condition for q = " + str(q))
|
" fails Borodzik condition for q = " + str(q))
|
||||||
|
|
||||||
if settings.print_matrices:
|
if settings.print_matrices:
|
||||||
self.print_matrices_for_borodzik(q)
|
self.print_matrices_for_borodzik(q)
|
||||||
|
|
||||||
for delta_prime, delta_prime_bases in self.naik_2_fulfilling:
|
for delta_prime, delta_prime_bases in self.naik_2_fulfilling:
|
||||||
print "\nResults for candidate delta_prime = " + str(delta_prime)
|
print("\nResults for candidate delta_prime = " + str(delta_prime))
|
||||||
for p, bases_for_p in delta_prime_bases:
|
for p, bases_for_p in delta_prime_bases:
|
||||||
print "Results for p = " + str(p)
|
print("Results for p = " + str(p))
|
||||||
for k, p_k_basis in enumerate(bases_for_p):
|
for k, p_k_basis in enumerate(bases_for_p):
|
||||||
self.print_borodzik_for_p_k_basis(p, k, p_k_basis, q)
|
self.print_borodzik_for_p_k_basis(p, k, p_k_basis, q)
|
||||||
print "%" * 200 + "\n" * 3
|
print("%" * 200 + "\n" * 3)
|
||||||
|
|
||||||
def print_matrices_for_borodzik(self, q):
|
def print_matrices_for_borodzik(self, q):
|
||||||
print "\n\nSeifert matrix A:"
|
print("\n\nSeifert matrix A:")
|
||||||
print str(self.seifert)
|
print(str(self.seifert))
|
||||||
print "\n\nA + A^T:"
|
print("\n\nA + A^T:")
|
||||||
print str(self.seifert + self.seifert.transpose())
|
print(str(self.seifert + self.seifert.transpose()))
|
||||||
print "\n\nC"
|
print("\n\nC")
|
||||||
print str(self.matrix_C)
|
print(str(self.matrix_C))
|
||||||
# print "\nE^(-1)"
|
# print("\nE^(-1)")
|
||||||
# print str(self.E_inverse)
|
# print(str(self.E_inverse))
|
||||||
print "\n\nD - diagonal"
|
print("\n\nD - diagonal")
|
||||||
print str(self.diagonal)
|
print(str(self.diagonal))
|
||||||
print "\n\nE"
|
print("\n\nE")
|
||||||
print str(self.matrix_E_inverse.inverse())
|
print(str(self.matrix_E_inverse.inverse()))
|
||||||
print "\n\nC^T * E^{-1} * D^{-1}"
|
print("\n\nC^T * E^{-1} * D^{-1}")
|
||||||
print self.get_C_tran_E_inv_D_inv()
|
print(self.get_C_tran_E_inv_D_inv())
|
||||||
|
|
||||||
def print_borodzik_for_p_k_basis(self, p, k, p_k_basis, q):
|
def print_borodzik_for_p_k_basis(self, p, k, p_k_basis, q):
|
||||||
|
|
||||||
@ -1188,73 +860,52 @@ class PeriodicityTester(object):
|
|||||||
P = p^(k + 1) * X.transpose() * self.get_C_tran_E_inv_D_inv() * X
|
P = p^(k + 1) * X.transpose() * self.get_C_tran_E_inv_D_inv() * X
|
||||||
P_det = P.determinant()
|
P_det = P.determinant()
|
||||||
if settings.print_matrices:
|
if settings.print_matrices:
|
||||||
print "\nsubmatrix:"
|
print("\nsubmatrix:")
|
||||||
print self.C_tran_E_inv_D_inv[-n:, -n:]
|
print(self.C_tran_E_inv_D_inv[-n:, -n:])
|
||||||
print "\nP\n" + str(P)
|
print("\nP\n" + str(P))
|
||||||
print "\ndet(P) = " + str(P_det)
|
print("\ndet(P) = " + str(P_det))
|
||||||
if mod(P_det, p).is_square():
|
if mod(P_det, p).is_square():
|
||||||
print ("det(P) % p = " + str(P_det % p) +
|
print("det(P) % p = " + str(P_det % p) +
|
||||||
" is a square => epsilon_1 := 1")
|
" is a square => epsilon_1 := 1")
|
||||||
epsilon_1 = 1
|
epsilon_1 = 1
|
||||||
else:
|
else:
|
||||||
print ("det(P) % p = " + str(P_det % p) +
|
print("det(P) % p = " + str(P_det % p) +
|
||||||
" isn't a square => episilon_1 := -1")
|
" isn't a square => episilon_1 := -1")
|
||||||
epsilon_1 = -1
|
epsilon_1 = -1
|
||||||
|
|
||||||
# p % 4 and n % 4, and epsilon_2
|
# p % 4 and n % 4, and epsilon_2
|
||||||
print "\np % 4 = " + str(p) + " % 4 = " + str(p % 4)
|
print("\np % 4 = " + str(p) + " % 4 = " + str(p % 4))
|
||||||
print "n % 4 = " + str(n) + " % 4 = " + str(n % 4)
|
print("n % 4 = " + str(n) + " % 4 = " + str(n % 4))
|
||||||
if p % 4 == 3 and n % 4 == 2:
|
if p % 4 == 3 and n % 4 == 2:
|
||||||
print "(p % 4 == 3 and n % 4 == 2) => episilon_2 := -1"
|
print("(p % 4 == 3 and n % 4 == 2) => episilon_2 := -1")
|
||||||
epsilon_2 = -1
|
epsilon_2 = -1
|
||||||
else:
|
else:
|
||||||
print "(p % 4 != 3 or n % 4 != 2) => episilon_2 := 1"
|
print("(p % 4 != 3 or n % 4 != 2) => episilon_2 := 1")
|
||||||
epsilon_2 = 1
|
epsilon_2 = 1
|
||||||
|
|
||||||
# epsilon and eta
|
# epsilon and eta
|
||||||
print "epsilon = epsilon_1 * epsilon_2 = " + str(epsilon_1 * epsilon_2)
|
print("epsilon = epsilon_1 * epsilon_2 = " + str(epsilon_1 * epsilon_2))
|
||||||
q_p = naik_number_dict[(q, p)]
|
q_p = naik_number_dict[(q, p)]
|
||||||
d = n / (2 * abs(q_p))
|
d = n / (2 * abs(q_p))
|
||||||
print "\nnaik_sign = " + str(sign(q_p))
|
print("\nnaik_sign = " + str(sign(q_p)))
|
||||||
print "eta = naik_sign^d = " + str(sign(q_p)^d)
|
print("eta = naik_sign^d = " + str(sign(q_p)^d))
|
||||||
if sign(q_p)^d == epsilon_1 * epsilon_2:
|
if sign(q_p)^d == epsilon_1 * epsilon_2:
|
||||||
print "eta == epsilon\n"
|
print("eta == epsilon\n")
|
||||||
else:
|
else:
|
||||||
print "eta != epsilon\n"
|
print("eta != epsilon\n")
|
||||||
|
|
||||||
|
|
||||||
class PrzytyckiTester(object):
|
class PrzytyckiTester(knots_periodicity.PrzytyckiTester):
|
||||||
|
|
||||||
def __init__(self, K, name, f_homfly_in=None):
|
def __init__(self, K, name, f_homfly_in=None):
|
||||||
|
super().__init__(self, K, name, f_homfly_in)
|
||||||
self.verbose = True
|
|
||||||
self.verbose = False
|
|
||||||
self.verbose = settings.debugging
|
self.verbose = settings.debugging
|
||||||
|
|
||||||
homflypt = self.get_homflypt_polynomial(K, name, f_homfly_in)
|
|
||||||
homfly_difference = homflypt(a, -z) - homflypt(a^-1, -z)
|
|
||||||
self.homfly_difference = z * homfly_difference
|
|
||||||
self.homflypt_polynomial = homflypt
|
|
||||||
|
|
||||||
if self.verbose:
|
if self.verbose:
|
||||||
print "\n" + "Knot " + name
|
print("\n" + "Knot " + name)
|
||||||
print "HOMFLYPT = " + str(homflypt)
|
print("HOMFLYPT = " + str(homflypt))
|
||||||
print ("HOMFLYPT(a, -z) - HOMFLYPT(a^-1, -z) = " +
|
print("HOMFLYPT(a, -z) - HOMFLYPT(a^-1, -z) = " +
|
||||||
str(homfly_difference))
|
str(homfly_difference))
|
||||||
print
|
print()
|
||||||
|
|
||||||
def get_homflypt_polynomial(self, K, name, f_homfly_in=None):
|
|
||||||
if f_homfly_in is not None:
|
|
||||||
try:
|
|
||||||
current_name, homflypt = f_homfly_in.readline().split(',')
|
|
||||||
while current_name != name:
|
|
||||||
current_name, homflypt = f_homfly_in.readline().split(',')
|
|
||||||
homflypt = sage_eval(homflypt, locals={'a': a, 'z': z})
|
|
||||||
return homflypt
|
|
||||||
except (AttributeError, ValueError) as e:
|
|
||||||
if self.verbose:
|
|
||||||
print "The file with HOMFLYPT is incorect!\n" + str(e)
|
|
||||||
return K.homfly_polynomial('a', 'z', 'lm')
|
|
||||||
|
|
||||||
def check_congruence(self, q):
|
def check_congruence(self, q):
|
||||||
for i in range(q + 1):
|
for i in range(q + 1):
|
||||||
@ -1263,17 +914,17 @@ class PrzytyckiTester(object):
|
|||||||
coefficient_modulo_ideal = z_coefficient.quo_rem(ideal)[1]
|
coefficient_modulo_ideal = z_coefficient.quo_rem(ideal)[1]
|
||||||
coefficient_modulo_q = coefficient_modulo_ideal.change_ring(GF(q))
|
coefficient_modulo_q = coefficient_modulo_ideal.change_ring(GF(q))
|
||||||
if self.verbose:
|
if self.verbose:
|
||||||
print "\nv_" + str(i) + " = " + str(z_coefficient)
|
print("\nv_" + str(i) + " = " + str(z_coefficient))
|
||||||
print ("v_" + str(i) + " mod (a + a^-1)^(q - i) = " +
|
print("v_" + str(i) + " mod (a + a^-1)^(q - i) = " +
|
||||||
str(coefficient_modulo_ideal))
|
str(coefficient_modulo_ideal))
|
||||||
print ("(v_" + str(i) + " mod (a + a^-1)^(q - i)) mod q = " +
|
print("(v_" + str(i) + " mod (a + a^-1)^(q - i)) mod q = " +
|
||||||
str(coefficient_modulo_q))
|
str(coefficient_modulo_q))
|
||||||
if coefficient_modulo_q != 0:
|
if coefficient_modulo_q != 0:
|
||||||
return 0
|
return 0
|
||||||
return 1
|
return 1
|
||||||
|
|
||||||
|
|
||||||
def check_criteria(name, pd_code, f_homfly_in=None):
|
def check_criteria(name, pd_code, A=None, f_homfly_in=None):
|
||||||
|
|
||||||
if settings.only_chosen and name not in settings.set_to_check:
|
if settings.only_chosen and name not in settings.set_to_check:
|
||||||
return None
|
return None
|
||||||
@ -1300,6 +951,14 @@ def check_criteria(name, pd_code, f_homfly_in=None):
|
|||||||
continue
|
continue
|
||||||
|
|
||||||
tester.check_criteria_for_period(q)
|
tester.check_criteria_for_period(q)
|
||||||
|
if settings.debugging:
|
||||||
|
print("\n" + "#" * 30 + " Calculations for knot " + tester.name +
|
||||||
|
" and q = " + str(q) + " " + "#" * 30 + "\n")
|
||||||
|
tester.print_data_for_murasugi(q)
|
||||||
|
tester.print_data_for_naik_1(q)
|
||||||
|
tester.print_data_for_naik_2(q)
|
||||||
|
tester.print_data_for_borodzik(q)
|
||||||
|
|
||||||
tester.results.append([q, tester.murasugi, tester.naik_1,
|
tester.results.append([q, tester.murasugi, tester.naik_1,
|
||||||
tester.naik_2, tester.borodzik,
|
tester.naik_2, tester.borodzik,
|
||||||
tester.przytycki])
|
tester.przytycki])
|
||||||
@ -1309,46 +968,6 @@ def check_criteria(name, pd_code, f_homfly_in=None):
|
|||||||
return tester
|
return tester
|
||||||
|
|
||||||
|
|
||||||
def get_naik_number(q, p):
|
|
||||||
'''
|
|
||||||
Calculate the smallest integer i = [q, p] such that p^i == +/-1 mod q.
|
|
||||||
Signum of i shows whether rest is -1 or 1
|
|
||||||
'''
|
|
||||||
if gcd(q, p) > 1:
|
|
||||||
return 0
|
|
||||||
p_power = p
|
|
||||||
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):
|
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:
|
with open(settings.f_pd_knot_11_15, 'r') as f:
|
||||||
line = f.readline()
|
line = f.readline()
|
||||||
@ -1359,7 +978,7 @@ def check_11_to_15(f_out, f_homfly_out=None, f_homfly_in=None):
|
|||||||
tester = check_criteria(name, pd_code, f_homfly_in)
|
tester = check_criteria(name, pd_code, f_homfly_in)
|
||||||
if tester is None:
|
if tester is None:
|
||||||
continue
|
continue
|
||||||
tester.save_results(f_out, f_homfly_out)
|
tester.save_results(f_out) #, f_homfly_out)
|
||||||
|
|
||||||
|
|
||||||
def check_up_to_10(f_out, f_homfly_out=None, f_homfly_in=None):
|
def check_up_to_10(f_out, f_homfly_out=None, f_homfly_in=None):
|
||||||
@ -1373,7 +992,7 @@ def check_up_to_10(f_out, f_homfly_out=None, f_homfly_in=None):
|
|||||||
tester = check_criteria(name, pd_code, f_homfly_in)
|
tester = check_criteria(name, pd_code, f_homfly_in)
|
||||||
if tester is None:
|
if tester is None:
|
||||||
continue
|
continue
|
||||||
tester.save_results(f_out, f_homfly_out)
|
tester.save_results(f_out) #, f_homfly_out)
|
||||||
|
|
||||||
|
|
||||||
def test_all(f_out, f_homfly_out=None, f_homfly_in=None):
|
def test_all(f_out, f_homfly_out=None, f_homfly_in=None):
|
||||||
@ -1385,14 +1004,7 @@ def test_all(f_out, f_homfly_out=None, f_homfly_in=None):
|
|||||||
check_11_to_15(f_out, f_homfly_out, f_homfly_in)
|
check_11_to_15(f_out, f_homfly_out, f_homfly_in)
|
||||||
|
|
||||||
|
|
||||||
if __name__ == '__main__':
|
def main():
|
||||||
|
|
||||||
settings = MySettings()
|
|
||||||
S.<a, z> = LaurentPolynomialRing(ZZ)
|
|
||||||
R.<t> = LaurentPolynomialRing(ZZ)
|
|
||||||
prime_numbers = Primes()
|
|
||||||
naik_number_dict = {}
|
|
||||||
|
|
||||||
if not os.path.isfile(settings.f_old_results) \
|
if not os.path.isfile(settings.f_old_results) \
|
||||||
or not settings.check_old_results:
|
or not settings.check_old_results:
|
||||||
settings.check_old_results = False
|
settings.check_old_results = False
|
||||||
@ -1412,7 +1024,7 @@ if __name__ == '__main__':
|
|||||||
else:
|
else:
|
||||||
with open(settings.f_results_out, 'w') as f_out:
|
with open(settings.f_results_out, 'w') as f_out:
|
||||||
test_all(f_out)
|
test_all(f_out)
|
||||||
sys.exit()
|
|
||||||
|
|
||||||
with open(settings.f_old_results, 'r') as f_old_results:
|
with open(settings.f_old_results, 'r') as f_old_results:
|
||||||
if settings.save_homfly and settings.input_file_with_homflypt:
|
if settings.save_homfly and settings.input_file_with_homflypt:
|
||||||
@ -1431,3 +1043,14 @@ if __name__ == '__main__':
|
|||||||
else:
|
else:
|
||||||
with open(settings.f_results_out, 'w') as f_out:
|
with open(settings.f_results_out, 'w') as f_out:
|
||||||
test_all(f_out)
|
test_all(f_out)
|
||||||
|
|
||||||
|
|
||||||
|
if __name__ == '__main__':
|
||||||
|
|
||||||
|
settings = MySettings()
|
||||||
|
S.<a, z> = LaurentPolynomialRing(ZZ)
|
||||||
|
R.<t> = LaurentPolynomialRing(ZZ)
|
||||||
|
prime_numbers = Primes()
|
||||||
|
naik_number_dict = knots_periodicity.naik_number_dict
|
||||||
|
if '__file__' in globals():
|
||||||
|
main()
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user