806 lines
30 KiB
Python
806 lines
30 KiB
Python
#!/usr/bin/python
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import collections
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import numpy as np
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import itertools as it
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import re
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class TorusCable(object):
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def __init__(self, knot_formula, k_vector=None, q_vector=None):
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# q_i = 2 * k_i + 1
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if k_vector is None:
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if q_vector is None:
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# TBD docstring
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print("Please give a list of k (k_vector) \
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or q values (q_vector).")
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return None
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else:
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k_vector = [(q - 1)/2 for q in q_vector]
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elif q_vector is None:
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q_vector = [2 * k + 1 for k in k_vector]
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self.knot_formula = knot_formula
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self.k_vector = k_vector
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self.q_vector = q_vector
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k = k_vector
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self.knot_sum = eval(knot_formula)
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self.knot_description = self.get_knot_descrption()
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self.__sigma_function = None
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self.__signature_as_function_of_theta = None
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def get_knot_descrption(self):
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description = ""
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for knot in self.knot_sum:
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if knot[0] < 0:
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description += "-"
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description += "T("
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for k in knot:
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description += "2, " + str(2 * abs(k) + 1) + "; "
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description = description[:-2] + ") # "
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return description[:-3]
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# searching for signature == 0
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def get_signature_as_function_of_theta(self, verbose=False):
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if self.__signature_as_function_of_theta is None:
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self.__signature_as_function_of_theta = \
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self.__get_signature_as_function_of_theta(verbose=verbose)
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return self.__signature_as_function_of_theta
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# searching for signature == 0
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def __get_signature_as_function_of_theta(self, **key_args):
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if 'verbose' in key_args:
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verbose_default = key_args['verbose']
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else:
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verbose_default = False
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def signature_as_function_of_theta(*thetas, **kwargs):
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verbose = verbose_default
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if 'verbose' in kwargs:
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verbose = kwargs['verbose']
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len_a = len(self.knot_sum)
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len_t = len(thetas)
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# call with no arguments
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if len_t == 0:
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return signature_as_function_of_theta(*(len_a * [0]))
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if len_t != len_a:
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msg = "This function takes exactly " + str(len_a) + \
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" arguments or no argument at all (" + str(len_t) + \
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" given)."
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raise TypeError(msg)
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sf = SignatureFunction()
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# for each cable knot in cable sum apply theta
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for i, knot in enumerate(self.knot_sum):
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try:
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ssf = get_signature_summand_as_theta_function(*knot)
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sf += ssf(thetas[i])
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# in case wrong theata value was given
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except ValueError as e:
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print("ValueError: " + str(e.args[0]) +\
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" Please change " + str(i + 1) + ". parameter.")
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return None
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if verbose:
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print()
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print(str(thetas))
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print(sf)
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return sf
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signature_as_function_of_theta.__doc__ =\
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signature_as_function_of_theta_docstring
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return signature_as_function_of_theta
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# searching for signature == 0
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def check_for_null_theta_combinations(self, verbose=False):
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list_of_good_vectors= []
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number_of_null_comb = 0
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f = self.get_signature_as_function_of_theta(verbose=verbose)
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range_list = [range(abs(knot[-1]) + 1) for knot in self.knot_sum]
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for theta_vector in it.product(*range_list):
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if f(*theta_vector, verbose=False).is_zero_everywhere():
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list_of_good_vectors.append(theta_vector)
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m = len([theta for theta in theta_vector if theta != 0])
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number_of_null_comb += 2^m
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return number_of_null_comb, list_of_good_vectors
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# searching for signature == 0
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def eval_cable_for_null_signature(self, print_results=False, verbose=False):
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# search for zero combinations
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number_of_all_comb = self.get_number_of_combinations_of_theta()
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result = self.check_for_null_theta_combinations(verbose=verbose)
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number_of_null_comb, list_of_good_vectors = result
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if print_results:
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print()
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print(self.knot_description)
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print("Zero cases: " + str(number_of_null_comb))
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print("All cases: " + str(number_of_all_comb))
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print("Zero theta combinations: ")
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for el in list_of_good_vectors:
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print(el)
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if number_of_null_comb^2 >= number_of_all_comb:
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return number_of_null_comb, number_of_all_comb
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return None
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# check sigma for all v = s * [a_1, a_2, a_3, a_4] for s in [1, q_4 - 1]
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def __is_sigma_for_vector_class_big(self, theta_vector):
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[a_1, a_2, a_3, a_4] = theta_vector
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q_4 = self.q_vector[3]
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for shift in range(1, q_4):
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shifted_theta = [(shift * a) % q_4 for a in
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[a_1, a_2, a_3, a_4]]
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sigma_v = self.__sigma_function(shifted_theta)
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if abs(sigma_v) > 5 + np.count_nonzero(shifted_theta):
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return True
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return False
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def __tmp_print_all_sigma_for_vector_class(self, theta_vector):
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print("\n")
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print(self.knot_description)
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print("vector = " + str(theta_vector))
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[a_1, a_2, a_3, a_4] = theta_vector
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q_4 = self.q_vector[3]
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for shift in range(1, q_4):
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shifted_theta = [(shift * a) % q_4 for a in
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[a_1, a_2, a_3, a_4]]
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print(str(shifted_theta) + "\t\t" + \
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str(self.__sigma_function(shifted_theta)))
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print("\n")
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def __tmp_get_max_sigma_for_vector_class(self, theta_vector):
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# print("\n")
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# print(self.knot_description)
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# print("vector = " + str(theta_vector))
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max_sigma = (theta_vector, 0)
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[a_1, a_2, a_3, a_4] = theta_vector
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q_4 = self.q_vector[3]
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for shift in range(1, q_4):
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shifted_theta = [(shift * a) % q_4 for a in
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[a_1, a_2, a_3, a_4]]
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sigma = self.__sigma_function(shifted_theta)
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if abs(sigma) > abs(max_sigma[1]):
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max_sigma = (shifted_theta, sigma)
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assert max_sigma[1] == 0, knot_description
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# print("\n" + self.knot_description + "\t" + str(max_sigma[0]) +\
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# "\t" + str(max_sigma[1]))
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return max_sigma[1]
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def is_sigma_for_vector_class_big(self, theta_vector):
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if self.__sigma_function is None:
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self.__sigma_function = self.__get_sigma_function()
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return self.__is_sigma_for_vector_class_big(theta_vector)
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def __get_sigma_function(self):
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k_1, k_2, k_3, k_4 = [abs(k) for k in self.k_vector]
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q_4 = 2 * k_4 + 1
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ksi = 1/q_4
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sigma_q_1 = get_untwisted_signature_function(k_1)
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sigma_q_2 = get_untwisted_signature_function(k_2)
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sigma_q_3 = get_untwisted_signature_function(k_3)
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def sigma_function(theta_vector, print_results=False):
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# "untwisted" part (Levine-Tristram signatures)
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a_1, a_2, a_3, a_4 = theta_vector
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untwisted_part = 2 * (sigma_q_2(ksi * a_1) -
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sigma_q_2(ksi * a_2) +
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sigma_q_3(ksi * a_3) -
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sigma_q_3(ksi * a_4) +
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sigma_q_1(ksi * a_1 * 2) -
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sigma_q_1(ksi * a_4 * 2))
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# "twisted" part
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tp = [0, 0, 0, 0]
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for i, a in enumerate(theta_vector):
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if a:
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tp[i] = -q_4 + 2 * a - 2 * (a^2/q_4)
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twisted_part = tp[0] - tp[1] + tp[2] - tp[3]
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# if print_results:
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# self.print_results_LT(theta_vector, untwisted_part)
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# self.print_results_LT(theta_vector, twisted_part)
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sigma_v = untwisted_part + twisted_part
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return sigma_v
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return sigma_function
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def print_results_LT(self, theta_vector, untwisted_part):
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knot_description = self.knot_description
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k_1, k_2, k_3, k_4 = [abs(k) for k in self.k_vector]
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a_1, a_2, a_3, a_4 = theta_vector
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q_4 = 2 * k_4 + 1
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ksi = 1/q_4
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sigma_q_1 = get_untwisted_signature_function(k_1)
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sigma_q_2 = get_untwisted_signature_function(k_2)
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sigma_q_3 = get_untwisted_signature_function(k_3)
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print("\n\nLevine-Tristram signatures for the cable sum: ")
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print(knot_description)
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print("and characters:\n" + str(theta_vector) + ",")
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print("ksi = " + str(ksi))
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print("\n\n2 * (sigma_q_2(ksi * a_1) + " + \
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"sigma_q_1(ksi * a_1 * 2) - " +\
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"sigma_q_2(ksi * a_2) + " +\
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"sigma_q_3(ksi * a_3) - " +\
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"sigma_q_3(ksi * a_4) - " +\
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"sigma_q_1(ksi * a_4 * 2))" +\
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\
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" = \n\n2 * (sigma_q_2(" + \
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str(ksi) + " * " + str(a_1) + \
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") + sigma_q_1(" + \
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str(ksi) + " * " + str(a_1) + " * 2" + \
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") - sigma_q_2(" + \
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str(ksi) + " * " + str(a_2) + \
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") + sigma_q_3(" + \
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str(ksi) + " * " + str(a_3) + \
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") - sigma_q_3(" + \
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str(ksi) + " * " + str(a_4) + \
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") - sigma_q_1(" + \
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str(ksi) + " * " + str(a_4) + " * 2)) " + \
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\
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" = \n\n2 * (sigma_q_2(" + \
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str(mod_one(ksi * a_1)) + \
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") + sigma_q_1(" + \
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str(mod_one(ksi * a_1 * 2)) + \
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") - sigma_q_2(" + \
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str(mod_one(ksi * a_2)) + \
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") + sigma_q_3(" + \
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str(mod_one(ksi * a_3)) + \
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") - sigma_q_3(" + \
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str(mod_one(ksi * a_4)) + \
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") - sigma_q_1(" + \
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str(mod_one(ksi * a_4 * 2)) + \
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\
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") = \n\n2 * ((" + \
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str(sigma_q_2(ksi * a_1)) + \
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") + (" + \
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str(sigma_q_1(ksi * a_1 * 2)) + \
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") - (" + \
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str(sigma_q_2(ksi * a_2)) + \
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") + (" + \
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str(sigma_q_3(ksi * a_3)) + \
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") - (" + \
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str(sigma_q_3(ksi * a_4)) + \
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") - (" + \
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str(sigma_q_1(ksi * a_4 * 2)) + ")) = " + \
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"\n\n2 * (" + \
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str(sigma_q_2(ksi * a_1) +
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sigma_q_1(ksi * a_1 * 2) -
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sigma_q_2(ksi * a_2) +
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sigma_q_3(ksi * a_3) -
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sigma_q_3(ksi * a_4) -
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sigma_q_1(ksi * a_4 * 2)) + \
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") = " + str(untwisted_part))
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print("\nSignatures:")
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print("\nq_1 = " + str(2 * k_1 + 1) + ": " + repr(sigma_q_1))
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print("\nq_2 = " + str(2 * k_2 + 1) + ": " + repr(sigma_q_2))
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print("\nq_3 = " + str(2 * k_3 + 1) + ": " + repr(sigma_q_3))
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def get_number_of_combinations_of_theta(self):
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number_of_combinations = 1
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for knot in self.knot_sum:
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number_of_combinations *= (2 * abs(knot[-1]) + 1)
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return number_of_combinations
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def print_results_sigma(self, theta_vector, twisted_part):
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a_1, a_2, a_3, a_4 = theta_vector
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knot_description = self.knot_description
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q_4 = self.q_vector[-1]
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print("\n\nSigma values for the cable sum: ")
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print(knot_description)
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print("and characters: " + str(v_theta))
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print("\nsigma(T_{2, q_4}, ksi_a) = " + \
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"-q + (2 * a * (q_4 - a)/q_4) " +\
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"= -q + 2 * a - 2 * a^2/q_4 if a != 0,\n\t\t\t" +\
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" = 0 if a == 0.")
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print("\nsigma(T_{2, q_4}, chi_a_1) = ", end="")
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if a_1:
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print("- (" + str(q_4) + ") + 2 * " + str(a_1) + " + " +\
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"- 2 * " + str(a_1^2) + "/" + str(q_4) + \
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" = " + str(tp[0]))
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else:
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print("0")
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print("\nsigma(T_{2, q_4}, chi_a_2) = ", end ="")
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if a_2:
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print("- (" + str(q_4) + ") + 2 * " + str(a_2) + " + " +\
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"- 2 * " + str(a_2^2) + "/" + str(q_4) + \
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" = " + str(tp[1]))
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else:
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print("0", end="")
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print("\nsigma(T_{2, q_4}, chi_a_3) = ", end="")
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if a_3:
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print("- (" + str(q_4) + ") + 2 * " + str(a_3) + " + " +\
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"- 2 * " + str(a_3^2) + "/" + str(q_4) + \
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" = " + str(tp[2]))
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else:
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print("0", end="")
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print("\nsigma(T_{2, q_4}, chi_a_4) = ", end="")
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if a_4:
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print("- (" + str(q_4) + ") + 2 * " + str(a_4) + " + " +\
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"- 2 * " + str(a_4^2) + "/" + str(q_4) + \
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" = " + str(tp[3]))
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else:
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print("0")
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print("\n\nsigma(T_{2, q_4}, chi_a_1) " + \
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"- sigma(T_{2, q_4}, chi_a_2) " + \
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"+ sigma(T_{2, q_4}, chi_a_3) " + \
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"- sigma(T_{2, q_4}, chi_a_4) =\n" + \
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"sigma(T_{2, q_4}, " + str(a_1) + \
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") - sigma(T_{2, q_4}, " + str(a_2) + \
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") + sigma(T_{2, q_4}, " + str(a_3) + \
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") - sigma(T_{2, q_4}, " + str(a_4) + ") = " + \
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str(tp[0] - tp[1] + tp[2] - tp[3]))
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# searching for sigma > 5 + #(v_i != 0)
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def calculate_sigma(self, theta_vector):
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if self.__sigma_function is None:
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self.__sigma_function = self.__get_sigma_function()
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return self.__sigma_function(theta_vector)
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# searching for sigma > 5 + #(v_i != 0)
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def __check_combinations_in_range(self, range_product):
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large_sigma_for_all_combinations = True
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bad_vectors = []
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good_vectors = []
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q_4 = self.q_vector[-1]
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for vector in range_product:
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a_1, a_2, a_3, a_4 = vector
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if (a_1^2 - a_2^2 + a_3^2 - a_4^2) % q_4:
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continue
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if all(a in [1, q_4 - 1] for a in vector):
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is_all_one = True
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else:
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is_all_one = False
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if self.__is_sigma_for_vector_class_big(vector):
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good_vectors.append(vector)
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# if is_all_one:
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# print("\nHURA" * 100)
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# print(self.knot_description)
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# self.__tmp_print_all_sigma_for_vector_class(vector)
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# pass
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else:
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if is_all_one:
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self.__tmp_get_max_sigma_for_vector_class(vector)
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bad_vectors.append(vector)
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#####################################################
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if len(bad_vectors) > 8:
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break
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####################################################
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large_sigma_for_all_combinations = False
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return good_vectors, bad_vectors
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# searching for sigma > 5 + #(v_i != 0)
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def check_combinations_in_range(self, range_product):
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if self.__sigma_function is None:
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self.__sigma_function = self.__get_sigma_function()
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return self.__check_combinations_in_range(range_product)
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# searching for sigma > 5 + #(v_i != 0)
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def __check_all_combinations_in_ranges(self, list_of_ranges,
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print_results=True):
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all_combinations_pass = True
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all_bad_vectors = []
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number_of_all_good_v = 0
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for i, range_product in enumerate(list_of_ranges):
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good_v, bad_v = self.__check_combinations_in_range(range_product)
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number_of_all_good_v += len(good_v)
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all_bad_vectors = list(it.chain(all_bad_vectors, bad_v))
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if bad_v:
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all_combinations_pass = False
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if len(all_bad_vectors) > 8:
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break
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# if print_results:
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# print("good : bad:\t " + str(len(good_v)) +\
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# " : " + str(len(bad_v)))
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# if i in [0, 4,]:
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# print()
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# if bad_v:
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# print(bad_v)
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if print_results:
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print("good : bad:\t " + str(number_of_all_good_v) +\
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" : " + str(len(all_bad_vectors)))
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if len(all_bad_vectors) < 8:
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print()
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print(all_bad_vectors)
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return all_combinations_pass
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# searching for sigma > 5 + #(v_i != 0)
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def eval_cable_for_large_sigma(self, list_of_ranges,
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print_results=False, verbose=False):
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if self.__sigma_function is None:
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self.__sigma_function = self.__get_sigma_function()
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if print_results:
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# print("\n\n")
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# print(100 * "*")
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# print("Searching for a large signature values for the cable sum: ")
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print(self.knot_description, end="\t\t\t")
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# print()
|
|
if self.__check_all_combinations_in_ranges(list_of_ranges,
|
|
print_results=print_results):
|
|
return True
|
|
return False
|
|
|
|
|
|
|
|
class SignatureFunction(object):
|
|
|
|
def __init__(self, values=None, counter=None):
|
|
# set values of signature jumps
|
|
if counter is None:
|
|
counter = collections.Counter()
|
|
if values is None:
|
|
values = []
|
|
assert all(x < 1 for x, y in values),\
|
|
"Signature function is defined on the interval [0, 1)."
|
|
counter = collections.Counter(dict(values))
|
|
self.cnt_signature_jumps = counter
|
|
|
|
def sum_of_absolute_values(self):
|
|
return sum([abs(i) for i in self.cnt_signature_jumps.values()])
|
|
|
|
def is_zero_everywhere(self):
|
|
return not any(self.cnt_signature_jumps.values())
|
|
|
|
def double_cover(self):
|
|
# to read values for t^2
|
|
new_data = []
|
|
for jump_arg, jump in self.cnt_signature_jumps.items():
|
|
new_data.append((jump_arg/2, jump))
|
|
new_data.append((1/2 + jump_arg/2, jump))
|
|
return SignatureFunction(values=new_data)
|
|
|
|
def square_root(self):
|
|
# to read values for t^(1/2)
|
|
new_data = []
|
|
for jump_arg, jump in self.cnt_signature_jumps.items():
|
|
if jump_arg < 1/2:
|
|
new_data.append((2 * jump_arg, jump))
|
|
return SignatureFunction(values=new_data)
|
|
|
|
def minus_square_root(self):
|
|
# to read values for t^(1/2)
|
|
counter = collections.Counter()
|
|
for jump_arg, jump in self.cnt_signature_jumps.items():
|
|
if jump_arg >= 1/2:
|
|
counter[mod_one(2 * jump_arg)] = jump
|
|
return SignatureFunction(counter=counter)
|
|
|
|
def __lshift__(self, shift):
|
|
# A shift of the signature functions corresponds to the rotation.
|
|
return self.__rshift__(-shift)
|
|
|
|
def __rshift__(self, shift):
|
|
new_data = []
|
|
for jump_arg, jump in self.cnt_signature_jumps.items():
|
|
new_data.append((mod_one(jump_arg + shift), jump))
|
|
return SignatureFunction(values=new_data)
|
|
|
|
def __neg__(self):
|
|
counter = collections.Counter()
|
|
counter.subtract(self.cnt_signature_jumps)
|
|
return SignatureFunction(counter=counter)
|
|
|
|
# TBD short
|
|
def __add__(self, other):
|
|
counter = copy(self.cnt_signature_jumps)
|
|
counter.update(other.cnt_signature_jumps)
|
|
return SignatureFunction(counter=counter)
|
|
|
|
def __eq__(self, other):
|
|
return self.cnt_signature_jumps == other.cnt_signature_jumps
|
|
|
|
def __sub__(self, other):
|
|
counter = copy(self.cnt_signature_jumps)
|
|
counter.subtract(other.cnt_signature_jumps)
|
|
return SignatureFunction(counter=counter)
|
|
|
|
def __str__(self):
|
|
result = ''.join([str(jump_arg) + ": " + str(jump) + "\n"
|
|
for jump_arg, jump in sorted(self.cnt_signature_jumps.items())])
|
|
return result
|
|
|
|
def __repr__(self):
|
|
result = ''.join([str(jump_arg) + ": " + str(jump) + ", "
|
|
for jump_arg, jump in sorted(self.cnt_signature_jumps.items())])
|
|
return result[:-2] + "."
|
|
|
|
def __call__(self, arg):
|
|
# Compute the value of the signature function at the point arg.
|
|
# This requires summing all signature jumps that occur before arg.
|
|
arg = mod_one(arg)
|
|
cnt = self.cnt_signature_jumps
|
|
before_arg = [jump for jump_arg, jump in cnt.items() if jump_arg < arg]
|
|
return 2 * sum(before_arg) + cnt[arg]
|
|
|
|
def mod_one(n):
|
|
return n - floor(n)
|
|
|
|
def get_untwisted_signature_function(j):
|
|
# return the signature function of the T_{2,2k+1} torus knot
|
|
k = abs(j)
|
|
w = ([((2 * a + 1)/(4 * k + 2), -1 * sgn(j)) for a in range(k)] +
|
|
[((2 * a + 1)/(4 * k + 2), 1 * sgn(j))
|
|
for a in range(k + 1, 2 * k + 1)])
|
|
return SignatureFunction(values=w)
|
|
|
|
|
|
def get_signature_summand_as_theta_function(*arg):
|
|
def get_signture_function(theta):
|
|
# TBD: another formula (for t^2) description
|
|
|
|
k_n = abs(arg[-1])
|
|
if theta > k_n:
|
|
msg = "k for the pattern in the cable is " + str(arg[-1]) + \
|
|
". Parameter theta should not be larger than abs(k)."
|
|
raise ValueError(msg)
|
|
|
|
# twisted part
|
|
cable_signature = get_blanchfield_for_pattern(arg[-1], theta)
|
|
|
|
# untwisted part
|
|
for i, k in enumerate(arg[:-1][::-1]):
|
|
ksi = 1/(2 * k_n + 1)
|
|
power = 2^i
|
|
a = get_untwisted_signature_function(k)
|
|
shift = theta * ksi * power
|
|
b = a >> shift
|
|
c = a << shift
|
|
for _ in range(i):
|
|
b = b.double_cover()
|
|
c = c.double_cover()
|
|
cable_signature += b + c
|
|
test = b - c
|
|
test2 = -c + b
|
|
assert test == test
|
|
return cable_signature
|
|
get_signture_function.__doc__ = get_signture_function_docsting
|
|
return get_signture_function
|
|
|
|
|
|
def get_blanchfield_for_pattern(k_n, theta):
|
|
if theta == 0:
|
|
a = get_untwisted_signature_function(k_n)
|
|
return a.square_root() + a.minus_square_root()
|
|
|
|
results = []
|
|
k = abs(k_n)
|
|
ksi = 1/(2 * k + 1)
|
|
|
|
# lambda_odd, i.e. (theta + e) % 2 != 0
|
|
for e in range(1, k + 1):
|
|
if (theta + e) % 2 != 0:
|
|
results.append((e * ksi, 1 * sgn(k_n)))
|
|
results.append((1 - e * ksi, -1 * sgn(k_n)))
|
|
|
|
# lambda_even
|
|
# print("normal")
|
|
for e in range(1, theta):
|
|
if (theta + e) % 2 == 0:
|
|
results.append((e * ksi, 1 * sgn(k_n)))
|
|
results.append((1 - e * ksi, -1 * sgn(k_n)))
|
|
# print("reversed")
|
|
for e in range(theta + 1, k + 1):
|
|
if (theta + e) % 2 != 0:
|
|
continue
|
|
results.append((e * ksi, -1 * sgn(k_n)))
|
|
results.append((1 - e * ksi, 1 * sgn(k_n)))
|
|
return SignatureFunction(values=results)
|
|
|
|
|
|
def get_signature_summand_as_theta_function(*arg):
|
|
def get_signture_function(theta):
|
|
# TBD: another formula (for t^2) description
|
|
|
|
k_n = abs(arg[-1])
|
|
if theta > k_n:
|
|
msg = "k for the pattern in the cable is " + str(arg[-1]) + \
|
|
". Parameter theta should not be larger than abs(k)."
|
|
raise ValueError(msg)
|
|
|
|
# twisted part
|
|
cable_signature = get_blanchfield_for_pattern(arg[-1], theta)
|
|
|
|
# untwisted part
|
|
for i, k in enumerate(arg[:-1][::-1]):
|
|
ksi = 1/(2 * k_n + 1)
|
|
power = 2^i
|
|
a = get_untwisted_signature_function(k)
|
|
shift = theta * ksi * power
|
|
b = a >> shift
|
|
c = a << shift
|
|
for _ in range(i):
|
|
b = b.double_cover()
|
|
c = c.double_cover()
|
|
cable_signature += b + c
|
|
test = b - c
|
|
test2 = -c + b
|
|
assert test == test
|
|
return cable_signature
|
|
get_signture_function.__doc__ = get_signture_function_docsting
|
|
return get_signture_function
|
|
|
|
|
|
def get_untwisted_signature_function(j):
|
|
# return the signature function of the T_{2,2k+1} torus knot
|
|
k = abs(j)
|
|
w = ([((2 * a + 1)/(4 * k + 2), -1 * sgn(j)) for a in range(k)] +
|
|
[((2 * a + 1)/(4 * k + 2), 1 * sgn(j))
|
|
for a in range(k + 1, 2 * k + 1)])
|
|
return SignatureFunction(values=w)
|
|
|
|
|
|
TorusCable.get_number_of_combinations_of_theta.__doc__ = \
|
|
"""
|
|
Arguments:
|
|
arbitrary number of lists of numbers, each list encodes a single cable
|
|
Return:
|
|
number of possible theta values combinations that could be applied
|
|
for a given cable sum,
|
|
i.e. the product of q_j for j = {1,.. n},
|
|
where n is a number of direct components in the cable sum,
|
|
and q_j is the last q parameter for the component (a single cable)
|
|
"""
|
|
|
|
TorusCable.get_knot_descrption.__doc__ = \
|
|
"""
|
|
Arguments:
|
|
arbitrary number of lists of numbers, each list encodes a single cable.
|
|
Examples:
|
|
sage: get_knot_descrption([1, 3], [2], [-1, -2], [-3])
|
|
'T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)'
|
|
"""
|
|
|
|
TorusCable.eval_cable_for_null_signature.__doc__ = \
|
|
"""
|
|
This function calculates all possible twisted signature functions for
|
|
a knot that is given as an argument. The knot should be encoded as a list
|
|
of its direct component. Each component schould be presented as a list
|
|
of integers. This integers correspond to the k - values in each component/
|
|
cable. If a component is a mirror image of a cable the minus sign should
|
|
be written before each number for this component. For example:
|
|
eval_cable_for_null_signature([[1, 8], [2], [-2, -8], [-2]])
|
|
eval_cable_for_null_signature([[1, 2], [-1, -2]])
|
|
|
|
sage: eval_cable_for_null_signature([[1, 3], [2], [-1, -2], [-3]])
|
|
|
|
T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)
|
|
Zero cases: 1
|
|
All cases: 1225
|
|
Zero theta combinations:
|
|
(0, 0, 0, 0)
|
|
|
|
sage:
|
|
The numbers given to the function eval_cable_for_null_signature
|
|
are k-values for each component/cable in a direct sum.
|
|
"""
|
|
|
|
TorusCable.get_signature_as_function_of_theta.__doc__ = \
|
|
"""
|
|
Function intended to construct signature function for a connected
|
|
sum of multiple cables with varying theta parameter values.
|
|
Accept arbitrary number of arguments (depending on number of cables in
|
|
connected sum).
|
|
Each argument should be given as list of integer representing
|
|
k - parameters for a cable: parameters k_i (i=1,.., n-1) for satelit knots
|
|
T(2, 2k_i + 1) and - the last one - k_n for a pattern knot T(2, 2k_n + 1).
|
|
Returns a function that will take theta vector as an argument and return
|
|
an object SignatureFunction.
|
|
|
|
To calculate signature function for a cable sum and a theta values vector,
|
|
use as below.
|
|
|
|
sage: signature_function_generator = get_signature_as_function_of_theta(
|
|
[1, 3], [2], [-1, -2], [-3])
|
|
sage: sf = signature_function_generator(2, 1, 2, 2)
|
|
sage: print(sf)
|
|
0: 0
|
|
5/42: 1
|
|
1/7: 0
|
|
1/5: -1
|
|
7/30: -1
|
|
2/5: 1
|
|
3/7: 0
|
|
13/30: -1
|
|
19/42: -1
|
|
23/42: 1
|
|
17/30: 1
|
|
4/7: 0
|
|
3/5: -1
|
|
23/30: 1
|
|
4/5: 1
|
|
6/7: 0
|
|
37/42: -1
|
|
|
|
Or like below.
|
|
sage: print(get_signature_as_function_of_theta([1, 3], [2], [-1, -2], [-3]
|
|
)(2, 1, 2, 2))
|
|
0: 0
|
|
1/7: 0
|
|
1/6: 0
|
|
1/5: -1
|
|
2/5: 1
|
|
3/7: 0
|
|
1/2: 0
|
|
4/7: 0
|
|
3/5: -1
|
|
4/5: 1
|
|
5/6: 0
|
|
6/7: 0
|
|
"""
|
|
|
|
SignatureFunction.__doc__ = \
|
|
"""
|
|
This simple class encodes twisted and untwisted signature functions
|
|
of knots. Since the signature function is entirely encoded by its signature
|
|
jump, the class stores only information about signature jumps
|
|
in a dictionary self.cnt_signature_jumps.
|
|
The dictionary stores data of the signature jump as a key/values pair,
|
|
where the key is the argument at which the functions jumps
|
|
and value encodes the value of the jump. Remember that we treat
|
|
signature functions as defined on the interval [0,1).
|
|
"""
|
|
|
|
get_signture_function_docsting = \
|
|
"""
|
|
This function returns SignatureFunction for previously defined single
|
|
cable T_(2, q) and a theta given as an argument.
|
|
The cable was defined by calling function
|
|
get_signature_summand_as_theta_function(*arg)
|
|
with the cable description as an argument.
|
|
It is an implementaion of the formula:
|
|
Bl_theta(K'_(2, d)) =
|
|
Bl_theta(T_2, d) + Bl(K')(ksi_l^(-theta) * t)
|
|
+ Bl(K')(ksi_l^theta * t)
|
|
"""
|
|
|
|
signature_as_function_of_theta_docstring = \
|
|
"""
|
|
Arguments:
|
|
|
|
Returns object of SignatureFunction class for a previously defined
|
|
connected sum of len(arg) cables.
|
|
Accept len(arg) arguments: for each cable one theta parameter.
|
|
If call with no arguments, all theta parameters are set to be 0.
|
|
"""
|
|
|
|
mod_one.__doc__ = \
|
|
"""
|
|
Argument:
|
|
a number
|
|
Return:
|
|
the fractional part of the argument
|
|
Examples:
|
|
sage: mod_one(9 + 3/4)
|
|
3/4
|
|
sage: mod_one(-9 + 3/4)
|
|
3/4
|
|
sage: mod_one(-3/4)
|
|
1/4
|
|
"""
|
|
|
|
get_blanchfield_for_pattern.__doc__ = \
|
|
"""
|
|
Arguments:
|
|
k_n: a number s.t. q_n = 2 * k_n + 1, where
|
|
T(2, q_n) is a pattern knot for a single cable from a cable sum
|
|
theta: twist/character for the cable (value form v vector)
|
|
Return:
|
|
SignatureFunction created for twisted signature function
|
|
for a given cable and theta/character
|
|
Based on:
|
|
Proposition 9.8. in Twisted Blanchfield Pairing
|
|
(https://arxiv.org/pdf/1809.08791.pdf)
|
|
"""
|
|
|
|
get_signature_summand_as_theta_function.__doc__ = \
|
|
"""
|
|
Argument:
|
|
n integers that encode a single cable, i.e.
|
|
values of q_i for T(2,q_0; 2,q_1; ... 2, q_n)
|
|
Return:
|
|
a function that returns SignatureFunction for this single cable
|
|
and a theta given as an argument
|
|
"""
|