894 lines
30 KiB
Python
894 lines
30 KiB
Python
#!/usr/bin/python
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# TBD: read about Factory Method, variable in docstring, sage documentation
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# move settings to sep file
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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_null_signature as shown below.
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sage: eval_cable_for_null_signature([[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_null_signature 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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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_3) # -T(2, q_1; 2, q_3) #
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# T(2, q_2; 2, q_3) # -T(2, q_0; 2, q_2; 2, q_3)
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In knot_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_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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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 numpy as np
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import re
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class Config(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_formula is a schema for knots which signature function
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# will be calculated
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self.knot_formula = "[[k[0], k[1], k[3]], [-k[1], -k[3]], \
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[k[2], k[3]], [-k[0], -k[2], -k[3]]]"
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# self.knot_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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# self.knot_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.limit = 3
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# self.verbose = True
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self.verbose = False
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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.signature_jumps.
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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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# set values of signature jumps
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self.signature_jumps = collections.defaultdict(int)
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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.signature_jumps[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.signature_jumps.values()])
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def is_zero_everywhere(self):
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return not any(self.signature_jumps.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.signature_jumps.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.signature_jumps.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.signature_jumps.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.signature_jumps.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.signature_jumps.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.signature_jumps.items():
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new_data.append(jump_arg, -jump)
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return SignatureFunction(new_data)
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# TBD short
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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.signature_jumps.items():
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new_data[jump_arg] = jump + self.signature_jumps.get(jump_arg, 0)
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for jump_arg, jump in self.signature_jumps.items():
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if jump_arg not in new_data.keys():
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new_data[jump_arg] = self.signature_jumps[jump_arg]
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new_signature_function.signature_jumps = 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.signature_jumps.items())])
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def __call__(self, arg):
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# Compute the value of the signature function at the point arg.
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# This requires summing all signature jumps that occur before arg.
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assert 0 <= arg and arg < 1
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val = 0
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for jump_arg, jump in self.signature_jumps.items():
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if jump_arg < arg:
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val += 2 * jump
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elif jump_arg == arg:
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val += jump
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a = self.sum_of_absolute_values()
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b = self.is_zero_everywhere()
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assert (a and not b) or (not a and b)
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return val
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def main(arg):
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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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search_for_large_signature_value(limit=new_limit)
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# search_for_null_signature_value(limit=new_limit)
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# searching for signture > 5 + #(v_i != 0)
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def search_for_large_signature_value(knot_formula=None,
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limit=None,
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verbose=None):
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if limit is None:
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limit = config.limit
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if knot_formula is None:
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knot_formula = config.knot_formula
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if verbose is None:
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vebose = config.verbose
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k_vector_size = extract_max(knot_formula) + 1
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limit = max(limit, k_vector_size)
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combinations = it.combinations(range(1, limit + 1), k_vector_size)
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P = Primes()
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with open(config.f_results, 'w') as f_results:
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for c in combinations:
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k = [(P.unrank(i) - 1)/2 for i in c]
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knot_sum = eval(knot_formula)
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if config.only_slice_candidates:
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if not (k[3] > 4 * k[2] and
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k[2] > 4 * k[1] and
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k[1] > 4 * k[0]):
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print "niu niu"
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continue
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result = eval_cable_for_large_signature(knot_sum,
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print_results=False)
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# if result is not None:
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# knot_description, large_comb, all_comb = result
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# line = (str(k) + ", " + str(all_comb) + ", " +
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# str(all_comb) + "\n")
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# f_results.write(line)
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# searching for signture > 5 + #(v_i != 0)
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def eval_cable_for_large_signature(knot_sum,
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print_results=True,
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verbose=None):
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if verbose is None:
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verbose = config.verbose
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if len(knot_sum) != 4:
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print "Wrong number of cable direct summands!"
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return None
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q = 2 * abs(knot_sum[-1][-1]) + 1
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f = get_function_of_theta_for_sum(*knot_sum, verbose=False)
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g = get_function_of_theta_for_sum_test(*knot_sum, verbose=False)
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knot_description = get_knot_descrption(*knot_sum)
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# large_value_combinations = 0
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# good_thetas_list = []
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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 "eval_cable_for_large_signature - knot_description: "
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# print knot_description
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print "\n\n"
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print knot_description
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for v_theta in it.product(*ranges_list):
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if (v_theta[0]^2 - v_theta[1]^2 + v_theta[2]^2 - v_theta[3]^2) % q:
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continue
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y = f(*v_theta)(1/2)
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j = g(*v_theta)(1/2)
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assert y == j
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if abs(y) > 5 + np.count_nonzero(v_theta):
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print "\nLarge signature value"
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else:
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print "\nSmall signature value"
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print knot_description
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print "v_theta: " + str(v_theta)
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condition = (str(v_theta[0]^2) + " - " + str(v_theta[1]^2) + " + " +
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str(v_theta[2]^2) + " - " + str(v_theta[3]^2))
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print condition
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print "non zero value in v_theta: " + str(np.count_nonzero(v_theta))
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print "signature at 1/2: " + str(y)
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# == 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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# searching for signature == 0
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def search_for_null_signature_value(knot_formula=None, limit=None):
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if limit is None:
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limit = config.limit
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if knot_formula is None:
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knot_formula = config.knot_formula
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k_vector_size = extract_max(knot_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(config.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 confi.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_formula)
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if is_trivial_combination(knot_sum):
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continue
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result = search_for_large_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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# searching for signature == 0
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def eval_cable_for_null_signature(knot_sum, print_results=True, verbose=None):
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# search for zero combinations
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if verbose is None:
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vebose = confi.verbose
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f = get_function_of_theta_for_sum(*knot_sum, verbose=False)
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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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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=False).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 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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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_test(*arg):
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sf = SignatureFunction([(0, 0)])
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def get_signture_function_test(theta):
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# untwisted part
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k_n = abs(arg[-1])
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cable_signature = sf
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# print k_0, k_1, k_2, k_3
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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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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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tp = get_blanchfield_for_pattern(arg[-1], theta)
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cable_signature += tp
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print "\ncs: "
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print cable_signature(1/2)
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tp_at = tp(1/2)
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print "tp: "
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print tp_at
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if theta:
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q_4 = 2 * k_n + 1
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alternativ = - q_4 + 2 * theta - (2 * theta^2)/q_4
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if arg[-1] < 0:
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alternativ = -alternativ
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else:
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alternativ = 0
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print "new tp: "
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print alternativ
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print float(alternativ)
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return cable_signature, tp_at, alternativ, 0
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get_signture_function_test.__doc__ = get_signture_function_docsting
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return get_signture_function_test
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def get_cable_signature_as_theta_function(*arg):
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def get_signture_function(theta):
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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
|
|
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
|
|
return cable_signature
|
|
get_signture_function.__doc__ = get_signture_function_docsting
|
|
return get_signture_function
|
|
|
|
|
|
def get_untwisted_signature_function(j):
|
|
"""This function returns 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(w)
|
|
|
|
|
|
def get_function_of_theta_for_sum_test(*arg, **key_args):
|
|
if 'verbose' in key_args:
|
|
verbose_default = key_args['verbose']
|
|
else:
|
|
verbose_default = confi.verbose
|
|
sf0 = SignatureFunction([(0, 0)])
|
|
sf0_test = SignatureFunction([(0, 0)])
|
|
|
|
|
|
def signature_function_for_sum_test(*thetas, **kwargs):
|
|
verbose = verbose_default
|
|
if 'verbose' in kwargs:
|
|
verbose = kwargs['verbose']
|
|
la = len(arg)
|
|
lt = len(thetas)
|
|
sf = sf0
|
|
sf_test = sf0_test
|
|
|
|
# call with no arguments
|
|
if lt == 0:
|
|
return signature_function_for_sum_test(*(la * [0]))
|
|
|
|
if lt != la:
|
|
msg = "This function takes exactly " + str(la) + \
|
|
" arguments or no argument at all (" + str(lt) + " given)."
|
|
raise TypeError(msg)
|
|
|
|
# for each cable in cable sum apply theta
|
|
twisted_part = 0
|
|
old_twisted_part = 0
|
|
# T(2, q_0; 2, q_1; 2, q_3) # -T(2, q_1; 2, q_3) #
|
|
# # T(2, q_2; 2, q_3) # -T(2, q_0; 2, q_2; 2, q_3)
|
|
k_1, k_2, k_4 = arg[0]
|
|
k_3 = arg[2][0]
|
|
ksi = 1/abs(2 * k_4 + 1)
|
|
print arg[0]
|
|
print str(k_1) + " " + str(k_2) + " " + str(k_3) + " " + str(k_4)
|
|
sigma_q_1 = get_untwisted_signature_function(k_1)
|
|
sigma_q_2 = get_untwisted_signature_function(k_2)
|
|
sigma_q_3 = get_untwisted_signature_function(k_3)
|
|
a_1, a_2, a_3, a_4 = thetas
|
|
untwisted_part = 2 * (sigma_q_2(ksi * a_1) +
|
|
sigma_q_1(ksi * a_1 * 2) -
|
|
sigma_q_2(ksi * a_2) +
|
|
sigma_q_3(ksi * a_3) -
|
|
sigma_q_3(ksi * a_4) -
|
|
sigma_q_1(ksi * a_4 * 2))
|
|
for i, knot in enumerate(arg):
|
|
try:
|
|
dssf, otp, tp, up = (get_cable_signature_as_theta_function_test(*knot))(thetas[i])
|
|
sf += dssf
|
|
twisted_part += tp
|
|
old_twisted_part += otp
|
|
# in case wrong theata value was given
|
|
except ValueError as e:
|
|
print "ValueError: " + str(e.args[0]) +\
|
|
" Please change " + str(i + 1) + ". parameter."
|
|
return None
|
|
print
|
|
print old_twisted_part
|
|
print twisted_part
|
|
print untwisted_part
|
|
|
|
if verbose:
|
|
print
|
|
print str(thetas)
|
|
print sf
|
|
return sf
|
|
signature_function_for_sum_test.__doc__ = signature_function_for_sum_docstring
|
|
return signature_function_for_sum_test
|
|
|
|
|
|
|
|
def get_function_of_theta_for_sum(*arg, **key_args):
|
|
if 'verbose' in key_args:
|
|
verbose_default = key_args['verbose']
|
|
else:
|
|
verbose_default = confi.verbose
|
|
def signature_function_for_sum(*thetas, **kwargs):
|
|
verbose = verbose_default
|
|
if 'verbose' in kwargs:
|
|
verbose = kwargs['verbose']
|
|
la = len(arg)
|
|
lt = len(thetas)
|
|
|
|
# call with no arguments
|
|
if lt == 0:
|
|
return signature_function_for_sum(*(la * [0]))
|
|
|
|
if lt != la:
|
|
msg = "This function takes exactly " + str(la) + \
|
|
" arguments or no argument at all (" + str(lt) + " given)."
|
|
raise TypeError(msg)
|
|
|
|
sf = SignatureFunction([(0, 0)])
|
|
|
|
# for each cable in cable sum apply theta
|
|
for i, knot in enumerate(arg):
|
|
try:
|
|
sf += (get_cable_signature_as_theta_function(*knot))(thetas[i])
|
|
# in case wrong theata value was given
|
|
except ValueError as e:
|
|
print "ValueError: " + str(e.args[0]) +\
|
|
" Please change " + str(i + 1) + ". parameter."
|
|
return None
|
|
if verbose:
|
|
print
|
|
print str(thetas)
|
|
print sf
|
|
return sf
|
|
signature_function_for_sum.__doc__ = signature_function_for_sum_docstring
|
|
return signature_function_for_sum
|
|
|
|
|
|
def get_number_of_combinations(*arg):
|
|
number_of_combinations = 1
|
|
for knot in arg:
|
|
number_of_combinations *= (2 * abs(knot[-1]) + 1)
|
|
return number_of_combinations
|
|
|
|
|
|
def extract_max(string):
|
|
numbers = re.findall('\d+', string)
|
|
numbers = map(int, numbers)
|
|
return max(numbers)
|
|
|
|
|
|
def mod_one(n):
|
|
return n - floor(n)
|
|
|
|
|
|
def get_knot_descrption(*arg):
|
|
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]
|
|
|
|
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_number_of_combinations.__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)
|
|
"""
|
|
|
|
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)'
|
|
"""
|
|
|
|
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
|
|
"""
|
|
|
|
search_for_null_signature_value.__doc__ = \
|
|
"""
|
|
This function calculates signature functions for knots constracted
|
|
accordinga a schema for a cable sum. The schema is given as an argument
|
|
or defined in the class Config.
|
|
Results of calculations will be writen to a file and the stdout.
|
|
limit is the upper bound for the first value in k_vector,
|
|
i.e k[0] value in a cable sum, where q_0 = 2 * k[0] + 1.
|
|
|
|
(the number of knots that will be constracted depends on limit value).
|
|
For each knot/cable sum the function eval_cable_for_null_signature is called.
|
|
eval_cable_for_null_signature calculetes the number of all possible thetas
|
|
(characters) and the number of combinations for which signature function
|
|
equeles zero. In case the first number is larger than squere of the second,
|
|
eval_cable_for_null_signature returns None (i.e. the knot can not be slice).
|
|
Data for knots that are candidates for slice knots are saved to a file.
|
|
"""
|
|
|
|
extract_max.__doc__ = \
|
|
"""
|
|
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
|
|
"""
|
|
|
|
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.
|
|
"""
|
|
|
|
get_function_of_theta_for_sum.__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_function_of_theta_for_sum(
|
|
[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_function_of_theta_for_sum([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_cable_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
|
|
"""
|
|
|
|
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_cable_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_function_for_sum_docstring = \
|
|
"""
|
|
Arguments:
|
|
|
|
Returns object of SignatureFunction class for a previously defined
|
|
connected sum of len(arg) cables.
|
|
Acept len(arg) arguments: for each cable one theta parameter.
|
|
If call with no arguments, all theta parameters are set to be 0.
|
|
"""
|
|
|
|
main.__doc__ = \
|
|
"""
|
|
This function is run if the script was called from the terminal.
|
|
It calls another function, search_for_null_signature_value,
|
|
to calculate signature functions for a schema
|
|
of a cable sum defined in the class Config.
|
|
Optionaly a parameter (a limit for k_0 value) can be given.
|
|
Thought to be run for time consuming calculations.
|
|
"""
|
|
config = Config()
|
|
if __name__ == '__main__' and '__file__' in globals():
|
|
# not called in interactive mode as __file__ is not defined
|
|
main(sys.argv)
|
|
|
|
|
|
# def calculate_form(x, y, q4):
|
|
# x1, x2, x3, x4 = x
|
|
# y1, y2, y3, y4 = y
|
|
# form = (x1 * y1 - x2 * y2 + x3 * y3 - x4 * y4) % q_4
|
|
# # TBD change for ring modulo q_4
|
|
# return form
|
|
#
|
|
# def check_condition(v, q4):
|
|
# if calculate_form(v, v, q4):
|
|
# return False
|
|
# return True
|
|
#
|
|
# def find_v(q4):
|
|
# results = []
|
|
# for i in range(q4):
|
|
# for j in range(q4):
|
|
# for k in range(q4):
|
|
# for m in range(q4):
|
|
# if check_condition([i, j, k, m], q_4):
|
|
# results.add(v)
|
|
# return results
|
|
#
|
|
# def check_inequality(q, v):
|
|
# a1, a2, a3, a4 = v
|
|
# q1, q2, q3, q4 = q
|
|
# pattern = [q1, q2, q4],[-q2, -q4],[q3, q4],[-q1, -q3, -q4]
|
|
# signature_function_generator = get_function_of_theta_for_sum(pattern)
|
|
# signature_function_for_sum = signature_function_generator(a1, a2, a3, a4)
|
|
#
|
|
# # sigma_v = sigma(q4, a1) - s(a2) + s(a3) - s(a4)
|
|
#
|
|
#
|