all calculations are implemented, while refactoring
This commit is contained in:
Maria Marchwicka
parent
a5ceb3fafa
commit
f9c9f5dd1d
@ -31,12 +31,12 @@ The numbers given to the function eval_cable_for_null_signature are k-values for
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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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get_signature_as_theta_function (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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T(2, q_1; 2, q_2; 2, q_4) # -T(2, q_2; 2, q_4) #
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# T(2, q_3; 2, q_4) # -T(2, q_1; 2, q_3; 2, q_4)
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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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@ -54,6 +54,7 @@ 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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import doc_signature
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class Config(object):
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@ -77,8 +78,8 @@ class Config(object):
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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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self.verbose = True
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# self.verbose = False
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class SignatureFunction(object):
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@ -95,6 +96,8 @@ class SignatureFunction(object):
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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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self.ttsignature_jumps = collections.Counter()
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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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@ -147,11 +150,16 @@ class SignatureFunction(object):
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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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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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print "\n" * 3
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print "other"
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print other.signature_jumps
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print "self"
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print self.signature_jumps
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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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@ -159,7 +167,28 @@ class SignatureFunction(object):
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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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tnew_signature_function = SignatureFunction()
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tnew_data = collections.defaultdict(int)
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self.ttsignature_jumps = collections.Counter(self.signature_jumps)
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other.ttsignature_jumps = collections.Counter(other.signature_jumps)
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for jump_arg, jump in other.ttsignature_jumps.items():
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tnew_data[jump_arg] = jump + self.ttsignature_jumps.get(jump_arg, 0)
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for jump_arg, jump in self.ttsignature_jumps.items():
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if jump_arg not in tnew_data.keys():
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tnew_data[jump_arg] = self.ttsignature_jumps[jump_arg]
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tt = other.ttsignature_jumps + self.ttsignature_jumps
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# tt = dict(tt)
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tt = collections.defaultdict(int, tt)
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print "\n" * 3
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print "tt"
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print tt
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print "new_data"
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print new_data
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assert new_data == tnew_data
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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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@ -232,68 +261,120 @@ def search_for_large_signature_value(knot_formula=None,
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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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knot_description = get_knot_descrption(*knot_sum)
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if verbose is None:
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verbose = config.verbose
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if verbose:
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print "\n\n"
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print 100 * "*"
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print "Searching for a large signature values for the cable sum: "
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print knot_description
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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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f = get_signature_as_theta_function(*knot_sum, verbose=False)
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# g = get_signature_as_theta_function_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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q = 2 * abs(knot_sum[-1][-1]) + 1
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q_4 = q
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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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theta_squers = [i^2 for i in v_theta]
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condition = "(" + str(theta_squers[0]) + " - " + str(theta_squers[1]) \
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+ " + " + str(theta_squers[1]) + " - " + \
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str(theta_squers[3]) + ") % " + str(q_4)
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if verbose:
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print "\nChecking for characters: " + str(v_theta)
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if (theta_squers[0] - theta_squers[1] +
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theta_squers[2] - theta_squers[3]) % q:
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if verbose:
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print "Condition not satisfied: " + str(condition) + " != 0."
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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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#
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#
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# twisted_part = 0
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# old_twisted_part = 0
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# # T(2, q_1; 2, q_2; 2, q_4) # -T(2, q_2; 2, q_4) #
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# # # T(2, q_3; 2, q_4) # -T(2, q_1; 2, q_3; 2, q_4)
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k_1, k_2, k_4 = [abs(i) for i in knot_sum[0]]
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k_3 = abs(knot_sum[2][1])
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q_4 = 2 * k_4 + 1
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ksi = 1/q_4
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print "k values: "
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print str(k_1) + " " + str(k_2) + " " + str(k_3) + " " + str(k_4)
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sigma_q_1 = get_untwisted_signature_function(k_1)
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sigma_q_2 = get_untwisted_signature_function(k_2)
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sigma_q_3 = get_untwisted_signature_function(k_3)
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a_1, a_2, a_3, a_4 = v_theta
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untwisted_part = 2 * (sigma_q_2(mod_one(ksi * a_1)) +
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sigma_q_1(mod_one(ksi * a_1 * 2)) -
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sigma_q_2(mod_one(ksi * a_2)) +
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sigma_q_3(mod_one(ksi * a_3)) -
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sigma_q_3(mod_one(ksi * a_4)) -
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sigma_q_1(mod_one(ksi * a_4 * 2)))
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# tp = [0, 0, 0, 0]
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# for i, a in enumerate(thetas):
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# if a:
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# tp[i] = -q_4 + 2 * a - (2 * a^2)/q_4
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# print "petla"
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# print i
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# print tp[i]
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# print 5 * "\n"
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# print tp
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# new_twisted_part = tp[0] - tp[1] + tp[2] - tp[3]
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# print new_twisted_part
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#
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# for i, knot in enumerate(arg):
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# try:
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# dssf = get_signature_summand_as_theta_function_test(*knot)(thetas[i])
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# sf += dssf
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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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# print "\nold_twisted_part"
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# print old_twisted_part
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# print "twisted_part: "
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# print new_twisted_part
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# print "untwisted_part: "
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# print untwisted_part
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# print "\n\n\n\n" + 50 * "*" + "\nsum " + str(untwisted_part + new_twisted_part)
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#
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#
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#
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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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print "\n\tLarge signature value"
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else:
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print "\nSmall signature value"
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print "\n\tSmall 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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@ -314,14 +395,14 @@ def search_for_null_signature_value(knot_formula=None, limit=None):
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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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if config.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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result = eval_cable_for_null_signature(knot_sum)
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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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@ -329,11 +410,11 @@ def search_for_null_signature_value(knot_formula=None, limit=None):
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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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def eval_cable_for_null_signature(knot_sum, print_results=False, 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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vebose = config.verbose
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f = get_signature_as_theta_function(*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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@ -416,7 +497,7 @@ def get_blanchfield_for_pattern(k_n, theta):
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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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def get_signature_summand_as_theta_function_test(*arg):
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sf = SignatureFunction([(0, 0)])
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@ -451,18 +532,8 @@ def get_cable_signature_as_theta_function_test(*arg):
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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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return cable_signature
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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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@ -472,7 +543,7 @@ def get_cable_signature_as_theta_function_test(*arg):
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def get_cable_signature_as_theta_function(*arg):
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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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@ -512,16 +583,16 @@ def get_untwisted_signature_function(j):
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return SignatureFunction(w)
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def get_function_of_theta_for_sum_test(*arg, **key_args):
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def get_signature_as_theta_function_test(*arg, **key_args):
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if 'verbose' in key_args:
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verbose_default = key_args['verbose']
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else:
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verbose_default = confi.verbose
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verbose_default = config.verbose
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sf0 = SignatureFunction([(0, 0)])
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sf0_test = SignatureFunction([(0, 0)])
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def signature_function_for_sum_test(*thetas, **kwargs):
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def signature_as_theta_function_test(*thetas, **kwargs):
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verbose = verbose_default
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if 'verbose' in kwargs:
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verbose = kwargs['verbose']
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@ -532,7 +603,7 @@ def get_function_of_theta_for_sum_test(*arg, **key_args):
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# call with no arguments
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if lt == 0:
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return signature_function_for_sum_test(*(la * [0]))
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return signature_as_theta_function_test(*(la * [0]))
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if lt != la:
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msg = "This function takes exactly " + str(la) + \
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@ -542,11 +613,11 @@ def get_function_of_theta_for_sum_test(*arg, **key_args):
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# for each cable in cable sum apply theta
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twisted_part = 0
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old_twisted_part = 0
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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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k_1, k_2, k_4 = arg[0]
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k_3 = arg[2][0]
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ksi = 1/abs(2 * k_4 + 1)
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# T(2, q_1; 2, q_2; 2, q_4) # -T(2, q_2; 2, q_4) #
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# # T(2, q_3; 2, q_4) # -T(2, q_1; 2, q_3; 2, q_4)
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k_1, k_2, k_4 = [abs(i) for i in arg[0]]
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k_3 = abs(arg[2][0])
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ksi = 1/(2 * k_4 + 1)
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print arg[0]
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print str(k_1) + " " + str(k_2) + " " + str(k_3) + " " + str(k_4)
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sigma_q_1 = get_untwisted_signature_function(k_1)
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@ -559,38 +630,54 @@ def get_function_of_theta_for_sum_test(*arg, **key_args):
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sigma_q_3(ksi * a_3) -
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sigma_q_3(ksi * a_4) -
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sigma_q_1(ksi * a_4 * 2))
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q_4 = 2 * k_4 + 1
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tp = [0, 0, 0, 0]
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for i, a in enumerate(thetas):
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if a:
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tp[i] = -q_4 + 2 * a - (2 * a^2)/q_4
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print "petla"
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print i
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print tp[i]
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print 5 * "\n"
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print tp
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new_twisted_part = tp[0] - tp[1] + tp[2] - tp[3]
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print new_twisted_part
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for i, knot in enumerate(arg):
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try:
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dssf, otp, tp, up = (get_cable_signature_as_theta_function_test(*knot))(thetas[i])
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dssf = get_signature_summand_as_theta_function_test(*knot)(thetas[i])
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sf += dssf
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twisted_part += tp
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old_twisted_part += otp
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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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print
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print "\nold_twisted_part"
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print old_twisted_part
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print twisted_part
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print "twisted_part: "
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print new_twisted_part
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print "untwisted_part: "
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print untwisted_part
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print "\n\n\n\n" + 50 * "*" + "\nsum " + str(untwisted_part + new_twisted_part)
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print "old sum at 1/2: "
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print sf(1/2)
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if verbose:
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print
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print str(thetas)
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print sf
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return sf
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signature_function_for_sum_test.__doc__ = signature_function_for_sum_docstring
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return signature_function_for_sum_test
|
||||
signature_as_theta_function_test.__doc__ = signature_as_theta_function_docstring
|
||||
return signature_as_theta_function_test
|
||||
|
||||
|
||||
|
||||
def get_function_of_theta_for_sum(*arg, **key_args):
|
||||
def get_signature_as_theta_function(*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_default = config.verbose
|
||||
def signature_as_theta_function(*thetas, **kwargs):
|
||||
verbose = verbose_default
|
||||
if 'verbose' in kwargs:
|
||||
verbose = kwargs['verbose']
|
||||
@ -599,7 +686,7 @@ def get_function_of_theta_for_sum(*arg, **key_args):
|
||||
|
||||
# call with no arguments
|
||||
if lt == 0:
|
||||
return signature_function_for_sum(*(la * [0]))
|
||||
return signature_as_theta_function(*(la * [0]))
|
||||
|
||||
if lt != la:
|
||||
msg = "This function takes exactly " + str(la) + \
|
||||
@ -611,7 +698,7 @@ def get_function_of_theta_for_sum(*arg, **key_args):
|
||||
# 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])
|
||||
sf += get_signature_summand_as_theta_function(*knot)(thetas[i])
|
||||
# in case wrong theata value was given
|
||||
except ValueError as e:
|
||||
print "ValueError: " + str(e.args[0]) +\
|
||||
@ -622,8 +709,8 @@ def get_function_of_theta_for_sum(*arg, **key_args):
|
||||
print str(thetas)
|
||||
print sf
|
||||
return sf
|
||||
signature_function_for_sum.__doc__ = signature_function_for_sum_docstring
|
||||
return signature_function_for_sum
|
||||
signature_as_theta_function.__doc__ = signature_as_theta_function_docstring
|
||||
return signature_as_theta_function
|
||||
|
||||
|
||||
def get_number_of_combinations(*arg):
|
||||
@ -757,7 +844,7 @@ eval_cable_for_null_signature.__doc__ = \
|
||||
component/cable in a direct sum.
|
||||
"""
|
||||
|
||||
get_function_of_theta_for_sum.__doc__ = \
|
||||
get_signature_as_theta_function.__doc__ = \
|
||||
"""
|
||||
Function intended to construct signature function for a connected
|
||||
sum of multiple cables with varying theta parameter values.
|
||||
@ -772,7 +859,7 @@ get_function_of_theta_for_sum.__doc__ = \
|
||||
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(
|
||||
sage: signature_function_generator = get_signature_as_theta_function(
|
||||
[1, 3], [2], [-1, -2], [-3])
|
||||
sage: sf = signature_function_generator(2, 1, 2, 2)
|
||||
sage: print sf
|
||||
@ -795,7 +882,7 @@ get_function_of_theta_for_sum.__doc__ = \
|
||||
37/42: -1
|
||||
|
||||
Or like below.
|
||||
sage: print get_function_of_theta_for_sum([1, 3], [2], [-1, -2], [-3]
|
||||
sage: print get_signature_as_theta_function([1, 3], [2], [-1, -2], [-3]
|
||||
)(2, 1, 2, 2)
|
||||
0: 0
|
||||
1/7: 0
|
||||
@ -811,7 +898,7 @@ get_function_of_theta_for_sum.__doc__ = \
|
||||
6/7: 0
|
||||
"""
|
||||
|
||||
get_cable_signature_as_theta_function.__doc__ = \
|
||||
get_signature_summand_as_theta_function.__doc__ = \
|
||||
"""
|
||||
Argument:
|
||||
n integers that encode a single cable, i.e.
|
||||
@ -826,7 +913,7 @@ 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)
|
||||
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)) =
|
||||
@ -834,7 +921,7 @@ get_signture_function_docsting = \
|
||||
+ Bl(K')(ksi_l^theta * t)
|
||||
"""
|
||||
|
||||
signature_function_for_sum_docstring = \
|
||||
signature_as_theta_function_docstring = \
|
||||
"""
|
||||
Arguments:
|
||||
|
||||
@ -853,41 +940,10 @@ main.__doc__ = \
|
||||
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)
|
||||
#
|
||||
#
|
||||
if __name__ == '__main__':
|
||||
global config
|
||||
config = Config()
|
||||
if '__file__' in globals():
|
||||
# skiped in interactive mode as __file__ is not defined
|
||||
main(sys.argv)
|
||||
|
||||
Loading…
Reference in New Issue
Block a user