From 737a556b8b83e6b4ae536bd575756393f59a8afc Mon Sep 17 00:00:00 2001 From: Maria Marchwicka Date: Mon, 20 Jan 2020 05:38:45 +0100 Subject: [PATCH] correction in Murasugi's criterion --- verbose_periodicity.sage | 93 +++++++++++++++++++++++++--------------- 1 file changed, 59 insertions(+), 34 deletions(-) diff --git a/verbose_periodicity.sage b/verbose_periodicity.sage index 17e81bd..154eea0 100644 --- a/verbose_periodicity.sage +++ b/verbose_periodicity.sage @@ -36,7 +36,9 @@ class MySettings(object): self.f_results_out = os.path.join(os.getcwd(), "results.out") self.f_old_results = os.path.join(os.getcwd(), "old_results.input") - self.periods = [3, 5, 7, 9, 11] + self.periods = [9] + + # self.periods = [3, 5, 7, 9, 11] self.set_to_check = self.get_set() # check only knots from defined set @@ -44,7 +46,7 @@ class MySettings(object): self.only_chosen = False self.debugging = True - self.debugging = False + # self.debugging = False # only if debugging self.print_matrices = True @@ -70,7 +72,7 @@ class MySettings(object): # self.input_file_with_homflypt = False self.check_old_results = True - self.check_old_results = False + # self.check_old_results = False if self.input_file_with_homflypt: if not os.path.isfile(self.f_homfly_lm_in): @@ -83,7 +85,7 @@ class MySettings(object): periodic_burde = set(["3_1", "5_1", "7_1", "8_19", "9_1", "9_35", "9_40", "9_41", "9_47", "9_49", "10_3", "10_123", "10_124"]) - # set_to_check |= periodic_burde + set_to_check |= periodic_burde # knots that fail Borodzik criterion self.fails_dict = { @@ -353,7 +355,7 @@ class MySettings(object): } set_to_check |= set(self.success_dict.keys()) - # knots that are known to be periodic + # knots that are known to be (not) periodic self.periods_dict = { "3_1": [3], "5_1": [5], @@ -392,7 +394,7 @@ class MySettings(object): "15a40549": [-5], "15a53966": [-5] } - # set_to_check |= set(self.periods_dict.keys()) + set_to_check |= set(self.periods_dict.keys()) # knots that have Alexander polynomial = 1 self.alexander_1 = set(["11n34", @@ -596,9 +598,11 @@ class MySettings(object): "15n163337", "15n165398", ]) - # set_to_check |= self.alexander_1 + set_to_check |= self.alexander_1 - set_to_check = set(["10_123"]) + set_to_check |= set(["10_123"]) + set_to_check |= set(["15n166130"]) + set_to_check = set(self.fails_dict.keys()) return set_to_check @@ -723,16 +727,17 @@ class PeriodicityTester(object): 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 + delta = delta_prime^q * (1 + t^1 + ... + t^(r-1))^(q-1) mod q_factor where "delta_prime" is a delta factor. ''' - quotient_delta = self.delta.change_ring(GF(q)) + 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)) + 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 @@ -754,18 +759,19 @@ class PeriodicityTester(object): 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: - f_q = naik_number_dict.setdefault((f, q), get_naik_number(f, q)) - if not (t_delta_dict[f] / (2 * f_q)).is_integer(): + 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(p, q) == 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*[p|q]. + 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: @@ -780,7 +786,7 @@ class PeriodicityTester(object): delta_prime_bases = [] maximum_in_diagonal = self.get_maximum_in_diagonal() for p in p_list: - p_q = naik_number_dict[(p, q)] + q_p = naik_number_dict[(q, p)] bases_for_p_torsion = [] factor_power = p # find all p^k torsion parts @@ -795,7 +801,7 @@ class PeriodicityTester(object): 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 * p_q)).is_integer(): + 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) @@ -812,6 +818,7 @@ class PeriodicityTester(object): 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] @@ -860,7 +867,7 @@ class PeriodicityTester(object): 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]). + 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): @@ -883,10 +890,10 @@ class PeriodicityTester(object): if not mod(P_det, p).is_square(): epsilon *= -1 # epsilon = epsilon_1 * epsilon_2 - p_q = naik_number_dict[(p, q)] - d = n / (2 * p_q) - # sign(p_q) - whether rest is -1 or 1 - if sign(p_q)^d != epsilon: + 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 @@ -1070,7 +1077,7 @@ class PeriodicityTester(object): if not p_list: print "List of factors was empty." for p in p_list: - g = abs(naik_number_dict[(p, q)]) + g = abs(naik_number_dict[(q, p)]) print "factor of del/del'(-1): " + str(p) print "Naik number: " + str(g) print "2 * Naik number:\t" + str(2 * g) @@ -1117,7 +1124,7 @@ class PeriodicityTester(object): for p, bases_for_p in delta_prime_bases: print "\nfactor p for delta prime:\t\t\t" + str(p) - g = abs(naik_number_dict[(p, q)]) + g = abs(naik_number_dict[(q, p)]) print "Naik number:\t\t" + str(g) print "2 * Naik number:\t" + str(2 * g) test_naik_number = p^g % q @@ -1210,11 +1217,11 @@ class PeriodicityTester(object): # epsilon and eta print "epsilon = epsilon_1 * epsilon_2 = " + str(epsilon_1 * epsilon_2) - p_q = naik_number_dict[(p, q)] - d = n / (2 * abs(p_q)) - print "\nnaik_sign = " + str(sign(p_q)) - print "eta = naik_sign^d = " + str(sign(p_q)^d) - if sign(p_q)^d == epsilon_1 * epsilon_2: + q_p = naik_number_dict[(q, p)] + d = n / (2 * abs(q_p)) + print "\nnaik_sign = " + str(sign(q_p)) + print "eta = naik_sign^d = " + str(sign(q_p)^d) + if sign(q_p)^d == epsilon_1 * epsilon_2: print "eta == epsilon\n" else: print "eta != epsilon\n" @@ -1306,9 +1313,9 @@ def check_criteria(name, pd_code, f_homfly_in=None): return tester -def get_naik_number(p, q): +def get_naik_number(q, p): ''' - Calculate the smallest integer i such that p^i == +/-1 mod q. + 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: @@ -1389,10 +1396,28 @@ if __name__ == '__main__': R. = LaurentPolynomialRing(ZZ) prime_numbers = Primes() naik_number_dict = {} - if not os.path.isfile(settings.f_old_results): - f = open(settings.f_old_results, 'w+') + + if not os.path.isfile(settings.f_old_results) \ + or not settings.check_old_results: settings.check_old_results = False - f.close() + if settings.save_homfly and settings.input_file_with_homflypt: + with open(settings.f_results_out, 'w') as f_out,\ + open(settings.f_homfly_lm_out, 'w') as f_homfly_out,\ + open(settings.f_homfly_lm_in, 'r') as f_homfly_in: + test_all(f_out, f_homfly_out, f_homfly_in) + elif settings.save_homfly: + with open(settings.f_results_out, 'w') as f_out,\ + open(settings.f_homfly_lm_out, 'w') as f_homfly_out: + test_all(f_out, f_homfly_out) + elif 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, None, f_homfly_in) + else: + with open(settings.f_results_out, 'w') as f_out: + test_all(f_out) + sys.exit() + with open(settings.f_old_results, 'r') as f_old_results: if settings.save_homfly and settings.input_file_with_homflypt: with open(settings.f_results_out, 'w') as f_out,\