classes as separate module - now working
This commit is contained in:
Maria Marchwicka
parent
0110e6161c
commit
d9986b63c1
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@ -1,6 +1,8 @@
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#!/usr/bin/python
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import collections
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import numpy as np
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import itertools as it
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import re
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class TorusCable(object):
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def __init__(self, knot_formula, k_vector=None, q_vector=None):
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@ -514,3 +516,290 @@ class SignatureFunction(object):
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def mod_one(n):
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return n - floor(n)
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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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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(values=w)
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def get_signature_summand_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
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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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test = b - c
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test2 = -c + b
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assert test == test
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return cable_signature
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get_signture_function.__doc__ = get_signture_function_docsting
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return get_signture_function
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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(values=results)
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def get_signature_summand_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
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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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test = b - c
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test2 = -c + b
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assert test == test
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return cable_signature
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get_signture_function.__doc__ = get_signture_function_docsting
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return get_signture_function
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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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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(values=w)
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TorusCable.get_number_of_combinations_of_theta.__doc__ = \
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"""
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Arguments:
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arbitrary number of lists of numbers, each list encodes a single cable
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Return:
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number of possible theta values combinations that could be applied
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for a given cable sum,
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i.e. the product of q_j for j = {1,.. n},
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where n is a number of direct components in the cable sum,
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and q_j is the last q parameter for the component (a single cable)
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"""
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TorusCable.get_knot_descrption.__doc__ = \
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"""
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Arguments:
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arbitrary number of lists of numbers, each list encodes a single cable.
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Examples:
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sage: get_knot_descrption([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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"""
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TorusCable.eval_cable_for_null_signature.__doc__ = \
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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_null_signature([[1, 8], [2], [-2, -8], [-2]])
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eval_cable_for_null_signature([[1, 2], [-1, -2]])
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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
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are k-values for each component/cable in a direct sum.
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"""
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TorusCable.get_signature_as_function_of_theta.__doc__ = \
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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_signature_as_function_of_theta(
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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_signature_as_function_of_theta([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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SignatureFunction.__doc__ = \
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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.cnt_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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get_signture_function_docsting = \
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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_signature_summand_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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signature_as_function_of_theta_docstring = \
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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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Accept 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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mod_one.__doc__ = \
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"""
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Argument:
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a number
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Return:
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the fractional part of the argument
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Examples:
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sage: mod_one(9 + 3/4)
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3/4
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sage: mod_one(-9 + 3/4)
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3/4
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sage: mod_one(-3/4)
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1/4
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"""
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get_blanchfield_for_pattern.__doc__ = \
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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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get_signature_summand_as_theta_function.__doc__ = \
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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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@ -11,6 +11,11 @@ import collections
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import itertools as it
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import numpy as np
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import re
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# try:
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# from cable_signature import SignatureFunction, TorusCable
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# except ModuleNotFoundError:
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os.system('sage --preparse cable_signature.sage')
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os.system('mv cable_signature.sage.py cable_signature.py')
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from cable_signature import SignatureFunction, TorusCable
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class Config(object):
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@ -189,7 +194,7 @@ def search_for_null_signature_value(knot_formula=None, limit=None):
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for k in combinations:
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if config.only_slice_candidates and k_vector_size == 5:
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k = get_shifted_combination(k)
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cable = TorusCable(knot_formula, k_vector=k)
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cable = TorusCable(knot_formula, k_vector=k)
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if is_trivial_combination(cable.knot_sum):
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print(cable.knot_sum)
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continue
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@ -201,6 +206,11 @@ def search_for_null_signature_value(knot_formula=None, limit=None):
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str(all_comb) + "\n")
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f_results.write(line)
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def extract_max(string):
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numbers = re.findall('\d+', string)
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numbers = map(int, numbers)
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return max(numbers)
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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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@ -209,144 +219,6 @@ def is_trivial_combination(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(values=results)
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def get_signature_summand_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
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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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test = b - c
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test2 = -c + b
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assert test == test
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return cable_signature
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get_signture_function.__doc__ = get_signture_function_docsting
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return get_signture_function
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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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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(values=w)
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def extract_max(string):
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numbers = re.findall('\d+', string)
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numbers = map(int, numbers)
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return max(numbers)
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def mod_one(n):
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return n - floor(n)
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get_blanchfield_for_pattern.__doc__ = \
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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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TorusCable.get_number_of_combinations_of_theta.__doc__ = \
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"""
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Arguments:
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arbitrary number of lists of numbers, each list encodes a single cable
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Return:
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number of possible theta values combinations that could be applied
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for a given cable sum,
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i.e. the product of q_j for j = {1,.. n},
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where n is a number of direct components in the cable sum,
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||||
and q_j is the last q parameter for the component (a single cable)
|
||||
"""
|
||||
|
||||
TorusCable.get_knot_descrption.__doc__ = \
|
||||
"""
|
||||
Arguments:
|
||||
arbitrary number of lists of numbers, each list encodes a single cable.
|
||||
Examples:
|
||||
sage: get_knot_descrption([1, 3], [2], [-1, -2], [-3])
|
||||
'T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)'
|
||||
"""
|
||||
|
||||
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__ = \
|
||||
"""
|
||||
|
|
@ -367,6 +239,17 @@ search_for_null_signature_value.__doc__ = \
|
|||
Data for knots that are candidates for slice knots are saved to a file.
|
||||
"""
|
||||
|
||||
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.
|
||||
"""
|
||||
|
||||
|
||||
extract_max.__doc__ = \
|
||||
"""
|
||||
Return:
|
||||
|
|
@ -378,136 +261,6 @@ extract_max.__doc__ = \
|
|||
3300
|
||||
"""
|
||||
|
||||
TorusCable.eval_cable_for_null_signature.__doc__ = \
|
||||
"""
|
||||
This function calculates all possible twisted signature functions for
|
||||
a knot that is given as an argument. The knot should be encoded as a list
|
||||
of its direct component. Each component schould be presented as a list
|
||||
of integers. This integers correspond to the k - values in each component/
|
||||
cable. If a component is a mirror image of a cable the minus sign should
|
||||
be written before each number for this component. For example:
|
||||
eval_cable_for_null_signature([[1, 8], [2], [-2, -8], [-2]])
|
||||
eval_cable_for_null_signature([[1, 2], [-1, -2]])
|
||||
|
||||
sage: eval_cable_for_null_signature([[1, 3], [2], [-1, -2], [-3]])
|
||||
|
||||
T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)
|
||||
Zero cases: 1
|
||||
All cases: 1225
|
||||
Zero theta combinations:
|
||||
(0, 0, 0, 0)
|
||||
|
||||
sage:
|
||||
The numbers given to the function eval_cable_for_null_signature
|
||||
are k-values for each component/cable in a direct sum.
|
||||
"""
|
||||
|
||||
TorusCable.get_signature_as_function_of_theta.__doc__ = \
|
||||
"""
|
||||
Function intended to construct signature function for a connected
|
||||
sum of multiple cables with varying theta parameter values.
|
||||
Accept arbitrary number of arguments (depending on number of cables in
|
||||
connected sum).
|
||||
Each argument should be given as list of integer representing
|
||||
k - parameters for a cable: parameters k_i (i=1,.., n-1) for satelit knots
|
||||
T(2, 2k_i + 1) and - the last one - k_n for a pattern knot T(2, 2k_n + 1).
|
||||
Returns a function that will take theta vector as an argument and return
|
||||
an object SignatureFunction.
|
||||
|
||||
To calculate signature function for a cable sum and a theta values vector,
|
||||
use as below.
|
||||
|
||||
sage: signature_function_generator = get_signature_as_function_of_theta(
|
||||
[1, 3], [2], [-1, -2], [-3])
|
||||
sage: sf = signature_function_generator(2, 1, 2, 2)
|
||||
sage: print(sf)
|
||||
0: 0
|
||||
5/42: 1
|
||||
1/7: 0
|
||||
1/5: -1
|
||||
7/30: -1
|
||||
2/5: 1
|
||||
3/7: 0
|
||||
13/30: -1
|
||||
19/42: -1
|
||||
23/42: 1
|
||||
17/30: 1
|
||||
4/7: 0
|
||||
3/5: -1
|
||||
23/30: 1
|
||||
4/5: 1
|
||||
6/7: 0
|
||||
37/42: -1
|
||||
|
||||
Or like below.
|
||||
sage: print(get_signature_as_function_of_theta([1, 3], [2], [-1, -2], [-3]
|
||||
)(2, 1, 2, 2))
|
||||
0: 0
|
||||
1/7: 0
|
||||
1/6: 0
|
||||
1/5: -1
|
||||
2/5: 1
|
||||
3/7: 0
|
||||
1/2: 0
|
||||
4/7: 0
|
||||
3/5: -1
|
||||
4/5: 1
|
||||
5/6: 0
|
||||
6/7: 0
|
||||
"""
|
||||
|
||||
get_signature_summand_as_theta_function.__doc__ = \
|
||||
"""
|
||||
Argument:
|
||||
n integers that encode a single cable, i.e.
|
||||
values of q_i for T(2,q_0; 2,q_1; ... 2, q_n)
|
||||
Return:
|
||||
a function that returns SignatureFunction for this single cable
|
||||
and a theta given as an argument
|
||||
"""
|
||||
SignatureFunction.__doc__ = \
|
||||
"""
|
||||
This simple class encodes twisted and untwisted signature functions
|
||||
of knots. Since the signature function is entirely encoded by its signature
|
||||
jump, the class stores only information about signature jumps
|
||||
in a dictionary self.cnt_signature_jumps.
|
||||
The dictionary stores data of the signature jump as a key/values pair,
|
||||
where the key is the argument at which the functions jumps
|
||||
and value encodes the value of the jump. Remember that we treat
|
||||
signature functions as defined on the interval [0,1).
|
||||
"""
|
||||
get_signture_function_docsting = \
|
||||
"""
|
||||
This function returns SignatureFunction for previously defined single
|
||||
cable T_(2, q) and a theta given as an argument.
|
||||
The cable was defined by calling function
|
||||
get_signature_summand_as_theta_function(*arg)
|
||||
with the cable description as an argument.
|
||||
It is an implementaion of the formula:
|
||||
Bl_theta(K'_(2, d)) =
|
||||
Bl_theta(T_2, d) + Bl(K')(ksi_l^(-theta) * t)
|
||||
+ Bl(K')(ksi_l^theta * t)
|
||||
"""
|
||||
|
||||
signature_as_function_of_theta_docstring = \
|
||||
"""
|
||||
Arguments:
|
||||
|
||||
Returns object of SignatureFunction class for a previously defined
|
||||
connected sum of len(arg) cables.
|
||||
Accept len(arg) arguments: for each cable one theta parameter.
|
||||
If call with no arguments, all theta parameters are set to be 0.
|
||||
"""
|
||||
|
||||
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.
|
||||
"""
|
||||
|
||||
if __name__ == '__main__':
|
||||
global config
|
||||
|
|
|
|||
Loading…
Reference in New Issue