628 lines
20 KiB
Python
628 lines
20 KiB
Python
#!/usr/bin/python
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# TBD: read about Factory Method, variable in docstring, sage documentation
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def calculate_form(x, y, q4):
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x1, x2, x3, x4 = x
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y1, y2, y3, y4 = y
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form = (x1 * y1 - x2 * y2 + x3 * y3 - x4 * y4) % q_4
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# TBD change for ring modulo q_4
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return form
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def check_condition(v, q4):
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if calculate_form(v, v, q4):
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return False
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return True
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def find_v(q4):
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results = []
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for i in range(q4):
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for j in range(q4):
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for k in range(q4):
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for m in range(q4):
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if check_condition([i, j, k, m], q_4):
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results.add(v)
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return results
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def check_inequality(q, v):
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a1, a2, a3, a4 = v
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q1, q2, q3, q4 = q
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pattern = [q1, q2, q4],[-q2, -q4],[q3, q4],[-q1, -q3, -q4]
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signature_function_generator = get_function_of_theta_for_sum(pattern)
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signature_function_for_sum = signature_function_generator(a1, a2, a3, a4)
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# sigma_v = sigma(q4, a1) - s(a2) + s(a3) - s(a4)
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"""
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This script calculates signature functions for knots (cable sums).
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The script can be run as a sage script from the terminal
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or used in interactive mode.
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A knot (cable sum) is encoded as a list where each element (also a list)
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corresponds to a cable knot, e.g. a list
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[[1, 3], [2], [-1, -2], [-3]] encodes
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T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7).
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To calculate the number of characters for which signature function vanish use
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the function eval_cable_for_thetas as shown below.
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sage: eval_cable_for_thetas([[1, 3], [2], [-1, -2], [-3]])
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T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)
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Zero cases: 1
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All cases: 1225
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Zero theta combinations:
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(0, 0, 0, 0)
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sage:
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The numbers given to the function eval_cable_for_thetas are k-values for each
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component/cable in a direct sum.
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To calculate signature function for a knot and a theta value, use function
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get_function_of_theta_for_sum (see help/docstring for details).
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"""
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import os
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import sys
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import collections
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import inspect
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import itertools as it
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import pandas as pd
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import re
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class MySettings(object):
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def __init__(self):
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self.f_results = os.path.join(os.getcwd(), "results.out")
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# is the ratio restriction for values in k_vector taken into account
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# False flag is usefull to make quick script tests
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self.only_slice_candidates = True
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# self.only_slice_candidates = False
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# knot_sum_formula is a schema for knots which signature function
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# will be calculated
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self.knot_sum_formula = "[[k[0], k[1], k[2]], [k[3], k[4]], \
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[-k[0], -k[3], -k[4]], [-k[1], -k[2]]]"
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"""
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About notation:
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Cables that we work with follow a schema:
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T(2, q_0; 2, q_1; 2, q_2) # T(2, q_1; 2, q_2) #
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# -T(2, q_3; 2, q_2) # -T(2, q_0; 2, q_3; 2, q_2)
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In knot_sum_formula each k[i] is related with some q_i value, where
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q_i = 2*k[i] + 1.
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So we can work in the following steps:
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1) choose a schema/formula by changing the value of knot_sum_formula
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2) set each q_i all or choose range in which q_i should varry
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3) choose vector v / theata vector.
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"""
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# self.knot_sum_formula = "[[k[0], k[1], k[2]], [k[3]],\
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# [-k[0], -k[1], -k[3]], [-k[2]]]"
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self.default_limit = 3
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class SignatureFunction(object):
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"""
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This simple class encodes twisted and untwisted signature functions
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of knots. Since the signature function is entirely encoded by its signature
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jump, the class stores only information about signature jumps
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in a dictionary self.data.
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The dictionary stores data of the signature jump as a key/values pair,
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where the key is the argument at which the functions jumps
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and value encodes the value of the jump. Remember that we treat
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signature functions as defined on the interval [0,1).
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"""
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def __init__(self, values=[]):
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# We will store data of signature jumps here.
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self.data = collections.defaultdict(int)
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# values contain initial data of singature jumps
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for jump_arg, jump in values:
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assert 0 <= jump_arg < 1, \
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"Signature function is defined on the interval [0, 1)."
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self.data[jump_arg] = jump
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def sum_of_absolute_values(self):
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return sum([abs(i) for i in self.data.values()])
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def double_cover(self):
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# to read values for t^2
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new_data = []
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for jump_arg, jump in self.data.items():
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new_data.append((mod_one(jump_arg/2), jump))
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new_data.append((mod_one(1/2 + jump_arg/2), jump))
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return SignatureFunction(new_data)
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def square_root(self):
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# to read values for t^(1/2)
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new_data = []
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for jump_arg, jump in self.data.items():
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if jump_arg < 1/2:
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new_data.append((2 * jump_arg, jump))
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return SignatureFunction(new_data)
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def get_signture_jump(self, t):
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return self.data.get(t, 0)
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def minus_square_root(self):
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# to read values for t^(1/2)
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new_data = []
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for jump_arg, jump in self.data.items():
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if jump_arg >= 1/2:
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new_data.append((mod_one(2 * jump_arg), jump))
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return SignatureFunction(new_data)
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def __lshift__(self, shift):
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# A shift of the signature functions corresponds to the rotation.
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return self.__rshift__(-shift)
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def __rshift__(self, shift):
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new_data = []
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for jump_arg, jump in self.data.items():
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new_data.append((mod_one(jump_arg + shift), jump))
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return SignatureFunction(new_data)
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def __neg__(self):
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# we can perform arithmetic operations on signature functions.
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new_data = []
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for jump_arg, jump in self.data.items():
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new_data.append(jump_arg, -jump)
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return SignatureFunction(new_data)
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def __add__(self, other):
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new_signature_function = SignatureFunction()
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new_data = collections.defaultdict(int)
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for jump_arg, jump in other.data.items():
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new_data[jump_arg] = jump + self.data.get(jump_arg, 0)
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for jump_arg, jump in self.data.items():
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if jump_arg not in new_data.keys():
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new_data[jump_arg] = self.data[jump_arg]
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new_signature_function.data = new_data
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return new_signature_function
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def __sub__(self, other):
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return self + other.__neg__()
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def __str__(self):
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return ''.join([str(jump_arg) + ": " + str(jump) + "\n"
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for jump_arg, jump in sorted(self.data.items())])
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def main(arg):
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"""
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This function is run if the script was called from the terminal.
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It calls another function, perform_calculations,
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to calculate signature functions for a schema
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of a cable sum defined in the class MySettings.
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Optionaly a parameter (a limit for k_0 value) can be given.
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Thought to be run for time consuming calculations.
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"""
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try:
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new_limit = int(arg[1])
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except:
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new_limit = None
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perform_calculations(limit=new_limit)
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def perform_calculations(knot_sum_formula=None, limit=None):
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"""
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This function calculates signature functions for knots constracted
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accordinga a schema for a cable sum. The schema is given as an argument
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or defined in the class MySettings.
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Results of calculations will be writen to a file and the stdout.
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limit is the upper bound for the first value in k_vector,
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i.e k[0] value in a cable sum, where q_0 = 2 * k[0] + 1.
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(the number of knots that will be constracted depends on limit value).
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For each knot/cable sum the function eval_cable_for_thetas is called.
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eval_cable_for_thetas calculetes the number of all possible thetas
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(characters) and the number of combinations for which signature function
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equeles zero. In case the first number is larger than squere of the second,
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eval_cable_for_thetas returns None (i.e. the knot can not be slice).
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Data for knots that are candidates for slice knots are saved to a file.
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"""
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settings = MySettings()
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if limit is None:
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limit = settings.default_limit
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if knot_sum_formula is None:
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knot_sum_formula = settings.knot_sum_formula
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k_vector_size = extract_max(knot_sum_formula) + 1
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combinations = it.combinations_with_replacement(range(1, limit + 1),
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k_vector_size)
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with open(settings.f_results, 'w') as f_results:
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for k in combinations:
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# print
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# print k
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# TBD: maybe the following condition or the function
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# get_shifted_combination should be redefined to a dynamic version
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if settings.only_slice_candidates and k_vector_size == 5:
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k = get_shifted_combination(k)
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# print k
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knot_sum = eval(knot_sum_formula)
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if is_trivial_combination(knot_sum):
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continue
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result = eval_cable_for_thetas(knot_sum, print_results=False)
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if result is not None:
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knot_description, null_comb, all_comb = result
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line = (str(k) + ", " + str(null_comb) + ", " +
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str(all_comb) + "\n")
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f_results.write(line)
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def is_trivial_combination(knot_sum):
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# for now is applicable only for schema that are sums of 4 cables
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if len(knot_sum) == 4:
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oposit_to_first = [-k for k in knot_sum[0]]
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if oposit_to_first in knot_sum:
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return True
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return False
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def get_shifted_combination(combination):
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# for now applicable only for schama
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# "[[k[0], k[1], k[2]], [k[3], k[4]],
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# [-k[0], -k[3], -k[4]], [-k[1], -k[2]]]"
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# shift the combination so that the knot can be a candidate for slice
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combination = [combination[0], 4 * combination[0] + combination[1],
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4 * (4 * combination[0] + combination[1]) + combination[2],
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4 * combination[0] + combination[3],
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4 * (4 * combination[0] + combination[3]) + combination[4]]
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return combination
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def get_blanchfield_for_pattern(k_n, theta):
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"""
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Arguments:
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k_n: a number s.t. q_n = 2 * k_n + 1, where
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T(2, q_n) is a pattern knot for a single cable from a cable sum
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theta: twist/character for the cable (value form v vector)
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Return:
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SignatureFunction created for twisted signature function
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for a given cable and theta/character
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Based on:
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Proposition 9.8. in Twisted Blanchfield Pairing
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(https://arxiv.org/pdf/1809.08791.pdf)
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"""
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if theta == 0:
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a = get_untwisted_signature_function(k_n)
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return a.square_root() + a.minus_square_root()
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results = []
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k = abs(k_n)
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ksi = 1/(2 * k + 1)
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# 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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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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# 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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continue
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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(results)
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def get_cable_signature_as_theta_function(*arg):
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"""
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Argument:
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n integers that encode a single cable, i.e.
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values of q_i for T(2,q_0; 2,q_1; ... 2, q_n)
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Return:
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a function that returns SignatureFunction for this single cable
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and a theta given as an argument
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"""
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def get_signture_function(theta):
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"""
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This function returns SignatureFunction for previously defined single
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cable T_(2, q) and a theta given as an argument.
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The cable was defined by calling function
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get_cable_signature_as_theta_function(*arg)
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with the cable description as an argument.
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It is an implementaion of the formula:
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Bl_theta(K'_(2, d)) =
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Bl_theta(T_2, d) + Bl(K')(ksi_l^(-theta) * t)
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+ Bl(K')(ksi_l^theta * t)
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"""
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# TBD: another formula (for t^2) description
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k_n = abs(arg[-1])
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if theta > k_n:
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msg = "k for the pattern in the cable is " + str(arg[-1]) + \
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". Parameter theta should not be larger than abs(k)."
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raise ValueError(msg)
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# twisted part
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cable_signature = get_blanchfield_for_pattern(arg[-1], theta)
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# untwisted part
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for i, k in enumerate(arg[:-1][::-1]):
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ksi = 1/(2 * k_n + 1)
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power = 2^i
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a = get_untwisted_signature_function(k)
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shift = theta * ksi * power
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b = a >> shift
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c = a << shift
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for _ in range(i):
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b = b.double_cover()
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c = c.double_cover()
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cable_signature += b + c
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return cable_signature
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return get_signture_function
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def get_untwisted_signature_function(j):
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"""This function returns the signature function of the T_{2,2k+1}
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torus knot."""
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k = abs(j)
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w = ([((2 * a + 1)/(4 * k + 2), -1 * sgn(j)) for a in range(k)] +
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[((2 * a + 1)/(4 * k + 2), 1 * sgn(j))
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for a in range(k + 1, 2 * k + 1)])
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return SignatureFunction(w)
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def get_function_of_theta_for_sum(*arg):
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"""
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Function intended to construct signature function for a connected
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sum of multiple cables with varying theta parameter values.
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Accept arbitrary number of arguments (depending on number of cables in
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connected sum).
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Each argument should be given as list of integer representing
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k - parameters for a cable: parameters k_i (i=1,.., n-1) for satelit knots
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T(2, 2k_i + 1) and - the last one - k_n for a pattern knot T(2, 2k_n + 1).
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Returns a function that will take theta vector as an argument and return
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an object SignatureFunction.
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To calculate signature function for a cable sum and a theta values vector,
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use as below.
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sage: signature_function_generator = get_function_of_theta_for_sum(
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[1, 3], [2], [-1, -2], [-3])
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sage: sf = signature_function_generator(2, 1, 2, 2)
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sage: print sf
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0: 0
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5/42: 1
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1/7: 0
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1/5: -1
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7/30: -1
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2/5: 1
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3/7: 0
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13/30: -1
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19/42: -1
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23/42: 1
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17/30: 1
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4/7: 0
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3/5: -1
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23/30: 1
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4/5: 1
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6/7: 0
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37/42: -1
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Or like below.
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sage: print get_function_of_theta_for_sum([1, 3], [2], [-1, -2], [-3]
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)(2, 1, 2, 2)
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0: 0
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1/7: 0
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1/6: 0
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1/5: -1
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2/5: 1
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3/7: 0
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1/2: 0
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4/7: 0
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3/5: -1
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4/5: 1
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5/6: 0
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6/7: 0
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"""
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def signature_function_for_sum(*thetas, **kwargs):
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"""
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Arguments:
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Returns object of SignatureFunction class for a previously defined
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connected sum of len(arg) cables.
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Acept len(arg) arguments: for each cable one theta parameter.
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If call with no arguments, all theta parameters are set to be 0.
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"""
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if 'verbose' in kwargs:
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verbose = kwargs['verbose']
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else:
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verbose = False
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la = len(arg)
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lt = len(thetas)
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# call with no arguments
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if lt == 0:
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return signature_function_for_sum(*(la * [0]))
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if lt != la:
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msg = "This function takes exactly " + str(la) + \
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" arguments or no argument at all (" + str(lt) + " given)."
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raise TypeError(msg)
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sf = SignatureFunction([(0, 0)])
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# for each cable in cable sum apply theta
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for i, knot in enumerate(arg):
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try:
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sf += (get_cable_signature_as_theta_function(*knot))(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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return signature_function_for_sum
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def eval_cable_for_thetas(knot_sum, print_results=True, verbose=False):
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"""
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This function calculates all possible twisted signature functions for
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a knot that is given as an argument. The knot should be encoded as a list
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of its direct component. Each component schould be presented as a list
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of integers. This integers correspond to the k - values in each component/
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cable. If a component is a mirror image of a cable the minus sign should
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be written before each number for this component. For example:
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eval_cable_for_thetas([[1, 8], [2], [-2, -8], [-2]])
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eval_cable_for_thetas([[1, 2], [-1, -2]])
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sage: eval_cable_for_thetas([[1, 3], [2], [-1, -2], [-3]])
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T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)
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Zero cases: 1
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All cases: 1225
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Zero theta combinations:
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(0, 0, 0, 0)
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sage:
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The numbers given to the function eval_cable_for_thetas are k-values for each
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component/cable in a direct sum.
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|
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"""
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f = get_function_of_theta_for_sum(*knot_sum)
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knot_description = get_knot_descrption(*knot_sum)
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all_combinations = get_number_of_combinations(*knot_sum)
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null_combinations = 0
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zero_theta_combinations = []
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|
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ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum]
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if verbose:
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print
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print knot_description
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for v_theta in it.product(*ranges_list):
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if f(*v_theta, verbose=verbose).sum_of_absolute_values() == 0:
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zero_theta_combinations.append(v_theta)
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m = len([theta for theta in v_theta if theta != 0])
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null_combinations += 2^m
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# else:
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# assert sum(v_theta) != 0
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|
|
|
if print_results:
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|
print
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|
print knot_description
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|
print "Zero cases: " + str(null_combinations)
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|
print "All cases: " + str(all_combinations)
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|
if zero_theta_combinations:
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|
print "Zero theta combinations: "
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|
for el in zero_theta_combinations:
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|
print el
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if null_combinations^2 >= all_combinations:
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|
return knot_description, null_combinations, all_combinations
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|
return None
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|
|
|
|
|
def check_squares(a, k):
|
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print
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|
p = 2 * k + 1
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|
k_0 = (p^2 - 1)/2
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|
knot_sum = [[a, k], [k_0], [-a, -k_0], [-k]]
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|
print get_knot_descrption(*knot_sum)
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|
if a * 4 >= p or is_trivial_combination(knot_sum):
|
|
if a * 4 >= p:
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|
print str(knot_sum)
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|
print "a * 4 >= p"
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|
else:
|
|
print "Trivial " + str(knot_sum)
|
|
return None
|
|
eval_cable_for_thetas(knot_sum)
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|
|
|
|
|
def get_number_of_combinations(*arg):
|
|
"""
|
|
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)
|
|
"""
|
|
number_of_combinations = 1
|
|
for knot in arg:
|
|
number_of_combinations *= (2 * abs(knot[-1]) + 1)
|
|
return number_of_combinations
|
|
|
|
|
|
def extract_max(string):
|
|
"""
|
|
Return:
|
|
maximum of absolute values of numbers from given string
|
|
Examples:
|
|
sage: extract_max("([1, 3], [2], [-1, -2], [-10])")
|
|
10
|
|
sage: extract_max("3, 55, ewewe, -42, 3300, 50")
|
|
3300
|
|
"""
|
|
numbers = re.findall('\d+', string)
|
|
numbers = map(int, numbers)
|
|
return max(numbers)
|
|
|
|
|
|
def mod_one(n):
|
|
"""
|
|
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
|
|
"""
|
|
return n - floor(n)
|
|
|
|
|
|
def get_knot_descrption(*arg):
|
|
"""
|
|
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)'
|
|
"""
|
|
description = ""
|
|
for knot in arg:
|
|
if knot[0] < 0:
|
|
description += "-"
|
|
description += "T("
|
|
for k in knot:
|
|
description += "2, " + str(2 * abs(k) + 1) + "; "
|
|
description = description[:-2] + ") # "
|
|
return description[:-3]
|
|
|
|
|
|
if __name__ == '__main__' and '__file__' in globals():
|
|
# not called in interactive mode as __file__ is not defined
|
|
main(sys.argv)
|