638 lines
21 KiB
Python
638 lines
21 KiB
Python
#!/usr/bin/python
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import numpy as np
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import itertools as it
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from typing import Iterable
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from collections import Counter
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from sage.arith.functions import LCM_list
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import warnings
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import re
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import matplotlib.pyplot as plt
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import inspect
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# 9.11 (9.8)
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# 9.15 (9.9)
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PLOTS_DIR = "plots"
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class CableSummand():
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def __init__(self, knot_as_k_values):
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self.knot_as_k_values = knot_as_k_values
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self._knot_description = self.get_summand_descrption(knot_as_k_values)
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self._signature_as_function_of_theta = None
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@staticmethod
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def get_summand_descrption(knot_as_k_values):
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description = ""
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if knot_as_k_values[0] < 0:
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description += "-"
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description += "T("
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for k in knot_as_k_values:
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description += "2, " + str(2 * abs(k) + 1) + "; "
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return description[:-2] + ")"
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@property
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def knot_description(self):
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return self._knot_description
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@property
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def signature_as_function_of_theta(self):
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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_summand_signature_as_theta_function()
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return self._signature_as_function_of_theta
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@classmethod
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def get_blanchfield_for_pattern(cls, k_n, theta=0):
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msg = "Theorem on which this function is based, assumes " +\
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"theta < k, where q = 2*k + 1 for pattern knot T(p, q)."
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if theta == 0:
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sf = cls.get_untwisted_signature_function(k_n)
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return sf.square_root() + sf.minus_square_root()
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k = abs(k_n)
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assert theta <= k, msg
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results = []
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ksi = 1/(2 * k + 1)
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counter = Counter()
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# print("lambda_odd, i.e. (theta + e) % 2 != 0")
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for e in range(1, k + 1):
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if (theta + e) % 2 != 0:
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counter[e * ksi] = 1 * sgn(k_n)
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counter[1 - e * ksi] = -1 * sgn(k_n)
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results.append((e * ksi, 1 * sgn(k_n)))
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results.append((1 - e * ksi, -1 * sgn(k_n)))
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# for example for k = 9 (q = 19) from this part we get
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# for even theta
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# 2/19: 1
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# 4/19: 1
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# 6/19: 1
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# 8/19: 1
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# 11/19: -1
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# 13/19: -1
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# 15/19: -1
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# 17/19: -1
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#
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# for odd theta
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# 1/19: 1
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# 3/19: 1
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# 5/19: 1
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# 7/19: 1
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# 9/19: 1
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# 10/19: -1
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# 12/19: -1
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# 14/19: -1
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# 16/19: -1
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# 18/19: -1
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# print("lambda_even")
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# print("normal")
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for e in range(1, theta):
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if (theta + e) % 2 == 0:
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results.append((e * ksi, 1 * sgn(k_n)))
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results.append((1 - e * ksi, -1 * sgn(k_n)))
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# print("reversed")
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for e in range(theta + 1, k + 1):
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if (theta + e) % 2 == 0:
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results.append((e * ksi, -1 * sgn(k_n)))
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results.append((1 - e * ksi, 1 * sgn(k_n)))
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return SignatureFunction(values=results)
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@classmethod
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def get_satellite_part(cls, *knot_as_k_values, theta=0):
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patt_k = knot_as_k_values[-1]
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ksi = 1/(2 * abs(patt_k) + 1)
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satellite_part = SignatureFunction()
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# For each knot summand consider k values in reversed order,
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# ommit k value for pattern.
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for layer_num, k in enumerate(knot_as_k_values[:-1][::-1]):
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sf = cls.get_untwisted_signature_function(k)
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shift = theta * ksi * 2^layer_num
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right_shift = sf >> shift
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left__shift = sf << shift
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for _ in range(layer_num):
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right_shift = right_shift.double_cover()
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left__shift = left__shift.double_cover()
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satellite_part += right_shift + left__shift
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return satellite_part
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@staticmethod
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def get_untwisted_signature_function(j):
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# return the signature function of the T_{2, 2k+1} torus knot
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k = abs(j)
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q = 2 * k + 1
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counter = Counter({(2 * a + 1)/(2 * q) : -sgn(j)
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for a in range(k)})
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counter.update(Counter({(2 * a + 1)/(2 * q) : sgn(j)
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for a in range(k + 1, q)}))
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return SignatureFunction(counter=counter)
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def get_summand_signature_as_theta_function(self):
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knot_as_k_values = self.knot_as_k_values
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def get_summand_signture_function(theta):
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patt_k = knot_as_k_values[-1]
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# theta should not be larger than k for the pattern.
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theta %= (2 * abs(patt_k) + 1)
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theta = min(theta, 2 * abs(patt_k) + 1 - theta)
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pattern_part = self.get_blanchfield_for_pattern(patt_k, theta)
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satellite_part = self.get_satellite_part(*knot_as_k_values,
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theta=theta)
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return pattern_part, satellite_part
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get_summand_signture_function.__doc__ = \
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get_summand_signture_function_docsting
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return get_summand_signture_function
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def get_file_name_for_summand_plot(self, theta=0):
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if self.knot_as_k_values[0] < 0:
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name = "inv_T_"
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else:
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name = "T_"
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for k in self.knot_as_k_values:
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name += str(abs(k)) + "_"
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name += "_theta_" + str(theta)
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return name
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def plot_summand_for_theta(self, theta, save_path=None):
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pp, sp = self.signature_as_function_of_theta(theta)
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title = self.knot_description + ", theta = " + str(theta)
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if save_path is not None:
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file_name = self.get_file_name_for_summand_plot(theta)
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save_path = os.path.join(save_path, file_name)
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pp.plot_sum_with_other(sp, title=title, save_path=save_path)
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def plot_summand(self):
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range_limit = min(self.knot_as_k_values[-1] + 1, 3)
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for theta in range(range_limit):
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self.plot_summand_for_theta(theta)
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class CableSum():
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def __init__(self, knot_sum):
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self.knot_sum_as_k_valus = knot_sum
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self.knot_summands = [CableSummand(k) for k in knot_sum]
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self.signature_as_function_of_theta = \
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self.get_signature_as_function_of_theta()
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def __call__(self, *thetas):
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return self.signature_as_function_of_theta(*thetas)
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def get_dir_name_for_plots(self, dir=None):
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dir_name = ''
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for knot in self.knot_summands:
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if knot.knot_as_k_values[0] < 0:
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dir_name += "inv_"
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dir_name += "T_"
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for k in knot.knot_as_k_values:
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k = 2 * abs (k) + 1
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dir_name += str(k) + "_"
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dir_name = dir_name[:-1]
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print(dir_name)
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dir_path = os.getcwd()
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if dir is not None:
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dir_path = os.path.join(dir_path, dir)
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dir_path = os.path.join(dir_path, dir_name)
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if not os.path.isdir(dir_path):
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os.mkdir(dir_path)
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return dir_name
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def plot_sum_for_theta_vector(self, thetas, save_to_dir=False):
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if save_to_dir:
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if not os.path.isdir(PLOTS_DIR):
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os.mkdir(PLOTS_DIR)
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dir_name = self.get_dir_name_for_plots(dir=PLOTS_DIR)
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save_path = os.path.join(os.getcwd(), PLOTS_DIR)
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save_path = os.path.join(save_path, dir_name)
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else:
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save_path = None
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for i, knot in enumerate(self.knot_summands):
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knot.plot_summand_for_theta(thetas[i], save_path=save_path)
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pp, sp = self.signature_as_function_of_theta(*thetas)
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title = self.knot_description + ", thetas = " + str(thetas)
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if save_path is not None:
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file_name = re.sub(r', ', '_', str(thetas))
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file_name = re.sub(r'[\[\]]', '', str(file_name))
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file_path = os.path.join(save_path, file_name)
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pp.plot_sum_with_other(sp, title=title, save_path=file_path)
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if save_path is not None:
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file_path = os.path.join(save_path, "all_" + file_name)
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sf_list = [knot.signature_as_function_of_theta(thetas[i])
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for i, knot in enumerate(self.knot_summands)]
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# pp, sp = knot.signature_as_function_of_theta(thetas[i])
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# (pp + sp) = sp.plot
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#
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# pp.plot_sum_with_other(sp, title=title, save_path=file_path)
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return dir_name
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def plot_all_summands(self):
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for knot in self.knot_summands:
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knot.plot_summand()
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@property
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def knot_description(self):
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return self._knot_description
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@property
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def patt_k_list(self):
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return self._patt_k_list
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@property
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def patt_q_list(self):
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return self._patt_q_list
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# q_order is LCM of all q values for pattern knots
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@property
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def q_order(self):
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return self._q_order
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@q_order.setter
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def q_order(self, val):
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self._q_order = val
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@property
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def knot_sum_as_k_valus(self):
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return self._knot_sum_as_k_valus
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@knot_sum_as_k_valus.setter
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def knot_sum_as_k_valus(self, knot_sum):
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self._knot_sum_as_k_valus = knot_sum
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self._knot_description = self.get_knot_descrption(knot_sum)
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self._patt_k_list = [abs(i[-1]) for i in knot_sum]
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self._patt_q_list = [2 * i + 1 for i in self._patt_k_list]
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if any(n not in Primes() for n in self._patt_q_list):
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msg = "Incorrect q-vector. This implementation assumes that" + \
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" all last q values are prime numbers.\n" + \
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str(self._patt_q_list)
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raise ValueError(msg)
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self.q_order = LCM_list(self._patt_q_list)
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def parse_thetas(self, *thetas):
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summands_num = len(self.knot_sum_as_k_valus)
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if not thetas:
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thetas = summands_num * (0,)
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elif len(thetas) == 1 and summands_num > 1:
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if isinstance(thetas[0], Iterable):
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if len(thetas[0]) >= summands_num:
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thetas = thetas[0]
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elif not thetas[0]:
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thetas = summands_num * (0,)
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elif thetas[0] == 0:
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thetas = summands_num * (0,)
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else:
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msg = "This function takes at least " + str(summands_num) + \
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" arguments or no argument at all (" + str(len(thetas)) \
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+ " given)."
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raise TypeError(msg)
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return tuple(thetas)
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@staticmethod
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def get_knot_descrption(knot_sum):
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description = ""
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for knot in 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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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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knot_desc = self.knot_description
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def signature_as_function_of_theta(*thetas, **kwargs):
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# print("\n\nsignature_as_function_of_theta " + knot_desc)
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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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thetas = self.parse_thetas(*thetas)
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satellite_part = SignatureFunction()
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pattern_part = SignatureFunction()
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# for each cable knot (summand) in cable sum apply theta
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for i, knot in enumerate(self.knot_summands):
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sfth = knot.signature_as_function_of_theta
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pp, sp = sfth(thetas[i])
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pattern_part += pp
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satellite_part += sp
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sf = pattern_part + satellite_part
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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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assert sf.total_sign_jump() == 0
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return pattern_part, satellite_part
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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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def is_metabolizer(self, theta):
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# Check if square alternating difference
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# divided by last q value is integer.
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result = sum(el^2 / self.patt_q_list[idx] * (-1)^idx
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for idx, el in enumerate(theta))
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# for idx, el in enumerate(theta):
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# old_sum += (el^2 / self.patt_q_list[idx] * (-1)^idx)
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return result.is_integer()
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def is_signature_big_in_ranges(self, ranges_list):
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for thetas in it.product(*ranges_list):
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# Check only non-zero metabolizers.
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if not self.is_metabolizer(thetas) or not any(thetas):
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continue
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signature_is_small = True
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# Check if any element generated by thetas vector
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# has a large signature.
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for shift in range(1, self.q_order):
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shifted_thetas = [shift * th for th in thetas]
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pp, sp = self.signature_as_function_of_theta(*shifted_thetas)
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sf = pp + sp
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limit = 5 + np.count_nonzero(shifted_thetas)
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extremum = abs(sf.extremum(limit=limit))
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if shift > 1:
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print(shifted_thetas, end=" ")
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print(extremum)
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if extremum > limit:
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signature_is_small = False
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break
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elif shift == 1:
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print("*" * 10)
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print(shifted_thetas, end=" ")
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print(extremum)
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if signature_is_small:
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print("\n" * 10 + "!" * 1000)
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return False
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return True
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def is_signature_big_for_all_metabolizers(self):
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num_of_summands = len(self.knot_sum_as_k_valus)
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if num_of_summands % 4:
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f_name = self.is_signature_big_for_all_metabolizers.__name__
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msg = "Function {}".format(f_name) + " is implemented only for " +\
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"knots that are direct sums of 4n direct summands."
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raise ValueError(msg)
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for shift in range(0, num_of_summands, 4):
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ranges_list = num_of_summands * [range(0, 1)]
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ranges_list[shift : shift + 3] = \
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[range(0, i + 1) for i in self.patt_k_list[shift: shift + 3]]
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ranges_list[shift + 3] = range(0, 2)
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if not self.is_signature_big_in_ranges(ranges_list):
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return False
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else:
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print("\nOK")
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return True
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class CableTemplate():
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def __init__(self, knot_formula, q_vector=None, k_vector=None,
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generate_q_vector=True, slice_candidate=True):
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self._knot_formula = knot_formula
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# q_i = 2 * k_i + 1
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if k_vector is not None:
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self.k_vector = k_vector
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elif q_vector is not None:
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self.q_vector = q_vector
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elif generate_q_vector:
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self.q_vector = self.get_q_vector(knot_formula, slice_candidate)
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@property
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def cable(self):
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if self._cable is None:
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msg = "q_vector for cable instance has not been set explicit. " + \
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"The variable is assigned a default value."
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warnings.warn(msg)
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self.fill_q_vector()
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return self._cable
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def fill_q_vector(self, q_vector=None, slice=True):
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if q_vector is None:
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q_vector = self.get_q_vector(self.knot_formula)
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self.q_vector = q_vector
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@property
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def knot_formula(self):
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return self._knot_formula
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@property
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def k_vector(self):
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return self._k_vector
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@k_vector.setter
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def k_vector(self, k):
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self._k_vector = k
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if self.extract_max(self.knot_formula) > len(k) - 1:
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msg = "The vector for knot_formula evaluation is to short!"
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msg += "\nk_vector " + str(k) + " \nknot_formula " \
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+ str(self.knot_formula)
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raise IndexError(msg)
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self.knot_sum_as_k_valus = eval(self.knot_formula)
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self._cable = CableSum(self.knot_sum_as_k_valus)
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self._q_vector = [2 * k_val + 1 for k_val in k]
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@property
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def q_vector(self):
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return self._q_vector
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@q_vector.setter
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def q_vector(self, new_q_vector):
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self.k_vector = [(q - 1)/2 for q in new_q_vector]
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@staticmethod
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def extract_max(string):
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numbers = re.findall(r'\d+', string)
|
|
numbers = map(int, numbers)
|
|
return max(numbers)
|
|
|
|
@classmethod
|
|
def get_q_vector(cls, knot_formula, slice=True):
|
|
lowest_number = 2
|
|
q_vector = [0] * (cls.extract_max(knot_formula) + 1)
|
|
P = Primes()
|
|
for layer in cls.get_layers_from_formula(knot_formula)[::-1]:
|
|
for el in layer:
|
|
q_vector[el] = P.next(lowest_number)
|
|
lowest_number = q_vector[el]
|
|
lowest_number *= 4
|
|
return q_vector
|
|
|
|
@staticmethod
|
|
def get_layers_from_formula(knot_formula):
|
|
k_indices = re.sub(r'[k-]', '', knot_formula)
|
|
k_indices = re.sub(r'\[\d+\]', lambda x: x.group()[1:-1], k_indices)
|
|
k_indices = eval(k_indices)
|
|
number_of_layers = max(len(lst) for lst in k_indices)
|
|
layers = []
|
|
for i in range(1, number_of_layers + 1):
|
|
layer = set()
|
|
for lst in k_indices:
|
|
if len(lst) >= i:
|
|
layer.add(lst[-i])
|
|
layers.append(layer)
|
|
return layers
|
|
|
|
def add_with_shift(self, other):
|
|
shift = self.extract_max(self.knot_formula) + 1
|
|
o_formula = re.sub(r'\d+', lambda x: str(int(x.group()) + shift),
|
|
other.knot_formula)
|
|
return self + CableTemplate(o_formula)
|
|
|
|
|
|
def __add__(self, other):
|
|
s_formula = self.knot_formula
|
|
o_formula = other.knot_formula
|
|
knot_formula = s_formula[:-1] + ",\n" + o_formula[1:]
|
|
cable_template = CableTemplate(knot_formula)
|
|
return cable_template
|
|
|
|
|
|
|
|
def mod_one(n):
|
|
return n - floor(n)
|
|
|
|
|
|
# CableSum.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)'
|
|
# """
|
|
|
|
CableSum.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
|
|
"""
|
|
|
|
get_summand_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_summand_signature_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.
|
|
"""
|
|
#
|
|
# CableSummand.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 pattern 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)
|
|
# """
|
|
|
|
# CableSummand.get_summand_signature_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
|
|
# """
|