correction in Murasugi's criterion
This commit is contained in:
Maria Marchwicka
parent
61e0669b3d
commit
737a556b8b
@ -36,7 +36,9 @@ class MySettings(object):
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self.f_results_out = os.path.join(os.getcwd(), "results.out")
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self.f_old_results = os.path.join(os.getcwd(), "old_results.input")
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self.periods = [3, 5, 7, 9, 11]
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self.periods = [9]
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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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@ -44,7 +46,7 @@ class MySettings(object):
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self.only_chosen = False
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self.debugging = True
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self.debugging = False
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# self.debugging = False
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# only if debugging
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self.print_matrices = True
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@ -70,7 +72,7 @@ class MySettings(object):
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# self.input_file_with_homflypt = False
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self.check_old_results = True
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self.check_old_results = False
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# self.check_old_results = 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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@ -83,7 +85,7 @@ class MySettings(object):
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periodic_burde = set(["3_1", "5_1", "7_1", "8_19", "9_1",
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"9_35", "9_40", "9_41", "9_47", "9_49",
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"10_3", "10_123", "10_124"])
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# set_to_check |= periodic_burde
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set_to_check |= periodic_burde
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# knots that fail Borodzik criterion
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self.fails_dict = {
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@ -353,7 +355,7 @@ class MySettings(object):
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}
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set_to_check |= set(self.success_dict.keys())
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# knots that are known to be periodic
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# knots that are known to be (not) periodic
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self.periods_dict = {
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"3_1": [3],
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"5_1": [5],
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@ -392,7 +394,7 @@ class MySettings(object):
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"15a40549": [-5],
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"15a53966": [-5]
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}
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# set_to_check |= set(self.periods_dict.keys())
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set_to_check |= set(self.periods_dict.keys())
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# knots that have Alexander polynomial = 1
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self.alexander_1 = set(["11n34",
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@ -596,9 +598,11 @@ class MySettings(object):
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"15n163337",
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"15n165398",
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])
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# set_to_check |= self.alexander_1
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set_to_check |= self.alexander_1
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set_to_check = set(["10_123"])
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set_to_check |= set(["10_123"])
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set_to_check |= set(["15n166130"])
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set_to_check = set(self.fails_dict.keys())
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return set_to_check
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@ -723,16 +727,17 @@ class PeriodicityTester(object):
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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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delta = delta_prime^q * (1 + t^1 + ... + t^(r-1))^(q-1) mod q_factor
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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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q_factor = factor(q)[0][0]
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quotient_delta = self.delta.change_ring(GF(q_factor))
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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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quotient_candidate = candidate.change_ring(GF(q_factor))
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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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@ -754,18 +759,19 @@ class PeriodicityTester(object):
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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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q_f = naik_number_dict.setdefault((q, f), get_naik_number(q, f))
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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 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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gcd(q, p) == 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*[q|p].
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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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# Proposition 2.7.
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delta_evaluated = self.delta(-1)
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for delta_prime, _ in self.murasugi_fulfilling:
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@ -780,7 +786,7 @@ class PeriodicityTester(object):
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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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q_p = naik_number_dict[(q, p)]
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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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@ -795,7 +801,7 @@ class PeriodicityTester(object):
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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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if not (len_non_zero / (2 * q_p)).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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@ -812,6 +818,7 @@ class PeriodicityTester(object):
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In particular naik_2 is set to be -1 if the criterion passes,
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but only in cases where P is an empty set.
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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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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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@ -860,7 +867,7 @@ class PeriodicityTester(object):
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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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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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@ -883,10 +890,10 @@ class PeriodicityTester(object):
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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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q_p = naik_number_dict[(q, p)]
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d = n / (2 * q_p)
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# sign(q_p) - whether rest is -1 or 1
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if sign(q_p)^d != epsilon:
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return False
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return True
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@ -1070,7 +1077,7 @@ class PeriodicityTester(object):
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if not p_list:
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print "List of factors was empty."
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for p in p_list:
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g = abs(naik_number_dict[(p, q)])
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g = abs(naik_number_dict[(q, p)])
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print "factor of del/del'(-1): " + str(p)
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print "Naik number: " + str(g)
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print "2 * Naik number:\t" + str(2 * g)
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@ -1117,7 +1124,7 @@ class PeriodicityTester(object):
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for p, bases_for_p in delta_prime_bases:
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print "\nfactor p for delta prime:\t\t\t" + str(p)
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g = abs(naik_number_dict[(p, q)])
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g = abs(naik_number_dict[(q, p)])
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print "Naik number:\t\t" + str(g)
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print "2 * Naik number:\t" + str(2 * g)
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test_naik_number = p^g % q
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@ -1210,11 +1217,11 @@ class PeriodicityTester(object):
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# epsilon and eta
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print "epsilon = epsilon_1 * epsilon_2 = " + str(epsilon_1 * epsilon_2)
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p_q = naik_number_dict[(p, q)]
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d = n / (2 * abs(p_q))
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print "\nnaik_sign = " + str(sign(p_q))
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print "eta = naik_sign^d = " + str(sign(p_q)^d)
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if sign(p_q)^d == epsilon_1 * epsilon_2:
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q_p = naik_number_dict[(q, p)]
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d = n / (2 * abs(q_p))
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print "\nnaik_sign = " + str(sign(q_p))
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print "eta = naik_sign^d = " + str(sign(q_p)^d)
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if sign(q_p)^d == epsilon_1 * epsilon_2:
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print "eta == epsilon\n"
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else:
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print "eta != epsilon\n"
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@ -1306,9 +1313,9 @@ def check_criteria(name, pd_code, f_homfly_in=None):
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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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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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'''
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if gcd(q, p) > 1:
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@ -1389,10 +1396,28 @@ if __name__ == '__main__':
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R.<t> = LaurentPolynomialRing(ZZ)
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prime_numbers = Primes()
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naik_number_dict = {}
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if not os.path.isfile(settings.f_old_results):
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f = open(settings.f_old_results, 'w+')
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if not os.path.isfile(settings.f_old_results) \
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or not settings.check_old_results:
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settings.check_old_results = False
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f.close()
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if settings.save_homfly and 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_out, 'w') as f_homfly_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_out, f_homfly_in)
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elif settings.save_homfly:
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with open(settings.f_results_out, 'w') as f_out,\
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open(settings.f_homfly_lm_out, 'w') as f_homfly_out:
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test_all(f_out, f_homfly_out)
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elif 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, None, 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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sys.exit()
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with open(settings.f_old_results, 'r') as f_old_results:
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if settings.save_homfly and settings.input_file_with_homflypt:
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with open(settings.f_results_out, 'w') as f_out,\
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