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Switch to Python 3. Dictionary is replaced by counter almost everywhere. Tests to each rpelacement.

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This commit is contained in:
Maria Marchwicka 2020-08-05 03:28:07 +02:00
parent c7c16ab4c0
commit 9b68822357
1 changed files with 200 additions and 130 deletions
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my_signature.sage
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@ -9,7 +9,6 @@ import sys
import collections import collections
import inspect import inspect
import itertools as it import itertools as it
import pandas as pd
import numpy as np import numpy as np
import re import re
@ -34,17 +33,17 @@ class Config(object):
self.verbose = False self.verbose = False
self.print_calculations_for_small_signature = True self.print_calculations_for_small_signature = True
# self.print_calculations_for_small_signature = False self.print_calculations_for_small_signature = False
self.print_calculations_for_large_signature = True self.print_calculations_for_large_signature = True
# self.print_calculations_for_large_signature = False self.print_calculations_for_large_signature = False
# is the ratio restriction for values in k_vector taken into account # is the ratio restriction for values in k_vector taken into account
# False flag is usefull to make quick script tests # False flag is usefull to make quick script tests
self.only_slice_candidates = True self.only_slice_candidates = True
self.only_slice_candidates = False # self.only_slice_candidates = False
self.stop_after_firts_large_signature = True self.stop_after_firts_large_signature = True
self.stop_after_firts_large_signature = False self.stop_after_firts_large_signature = False
@ -64,29 +63,45 @@ class SignatureFunction(object):
def __init__(self, values=[], counter=collections.Counter()): def __init__(self, values=[], counter=collections.Counter()):
# set values of signature jumps # set values of signature jumps
self.signature_jumps = collections.defaultdict(int, counter) self.signature_jumps = collections.defaultdict(int, counter)
self.counter_signature_jumps = counter self.cnt_signature_jumps = counter
if not counter: if not counter:
for jump_arg, jump in values: for jump_arg, jump in values:
assert 0 <= jump_arg < 1, \ assert 0 <= jump_arg < 1, \
"Signature function is defined on the interval [0, 1)." "Signature function is defined on the interval [0, 1)."
self.signature_jumps[jump_arg] = jump self.signature_jumps[jump_arg] = jump
self.counter_signature_jumps = collections.Counter(self.signature_jumps) self.cnt_signature_jumps = collections.Counter(self.signature_jumps)
def sum_of_absolute_values(self): def sum_of_absolute_values(self):
return sum([abs(i) for i in self.signature_jumps.values()]) result = sum([abs(i) for i in self.signature_jumps.values()])
test = sum([abs(i) for i in self.cnt_signature_jumps.values()])
assert test == result
return result
def is_zero_everywhere(self): def is_zero_everywhere(self):
result = not any(self.signature_jumps.values()) result = not any(self.signature_jumps.values())
assert result == (not any(self.counter_signature_jumps.values())) assert result == (not any(self.cnt_signature_jumps.values()))
if self.sum_of_absolute_values():
assert result == False
else:
assert result == True
return result return result
def double_cover(self): def double_cover(self):
# to read values for t^2 # to read values for t^2
new_data = [] new_data = []
for jump_arg, jump in self.signature_jumps.items(): for jump_arg, jump in self.signature_jumps.items():
new_data.append((mod_one(jump_arg/2), jump)) new_data.append((jump_arg/2, jump))
new_data.append((mod_one(1/2 + jump_arg/2), jump)) new_data.append((1/2 + jump_arg/2, jump))
return SignatureFunction(new_data)
t_data = []
for jump_arg, jump in self.cnt_signature_jumps.items():
t_data.append((jump_arg/2, jump))
t_data.append((1/2 + jump_arg/2, jump))
sf = SignatureFunction(t_data)
a = SignatureFunction(new_data)
assert a == sf
return sf
def square_root(self): def square_root(self):
# to read values for t^(1/2) # to read values for t^(1/2)
@ -94,34 +109,58 @@ class SignatureFunction(object):
for jump_arg, jump in self.signature_jumps.items(): for jump_arg, jump in self.signature_jumps.items():
if jump_arg < 1/2: if jump_arg < 1/2:
new_data.append((2 * jump_arg, jump)) new_data.append((2 * jump_arg, jump))
return SignatureFunction(new_data)
t_data = []
for jump_arg, jump in self.cnt_signature_jumps.items():
if jump_arg < 1/2:
t_data.append((2 * jump_arg, jump))
sf = SignatureFunction(t_data)
a = SignatureFunction(new_data)
assert a == sf
return sf
def minus_square_root(self): def minus_square_root(self):
# to read values for t^(1/2) # to read values for t^(1/2)
new_data = [] new_data = []
for jump_arg, jump in self.signature_jumps.items(): for jump_arg, jump in self.cnt_signature_jumps.items():
if jump_arg >= 1/2: if jump_arg >= 1/2:
new_data.append((mod_one(2 * jump_arg), jump)) new_data.append((mod_one(2 * jump_arg), jump))
return SignatureFunction(new_data) t_data = []
for jump_arg, jump in self.signature_jumps.items():
if jump_arg >= 1/2:
t_data.append((mod_one(2 * jump_arg), jump))
a = SignatureFunction(t_data)
sf = SignatureFunction(new_data)
assert a == sf
return sf
def __lshift__(self, shift): def __lshift__(self, shift):
# A shift of the signature functions corresponds to the rotation. # A shift of the signature functions corresponds to the rotation.
return self.__rshift__(-shift) return self.__rshift__(-shift)
def __rshift__(self, shift): def __rshift__(self, shift):
new_data = [] t_data = []
for jump_arg, jump in self.signature_jumps.items(): for jump_arg, jump in self.signature_jumps.items():
t_data.append((mod_one(jump_arg + shift), jump))
new_data = []
for jump_arg, jump in self.cnt_signature_jumps.items():
new_data.append((mod_one(jump_arg + shift), jump)) new_data.append((mod_one(jump_arg + shift), jump))
return SignatureFunction(new_data) sf = SignatureFunction(new_data)
a = SignatureFunction(t_data)
assert a == sf
return sf
def __neg__(self): def __neg__(self):
# we can perform arithmetic operations on signature functions.
new_data = [] new_data = []
for jump_arg, jump in self.signature_jumps.items(): for jump_arg, jump in self.signature_jumps.items():
new_data.append((jump_arg, -jump)) new_data.append((jump_arg, -jump))
sf = SignatureFunction(new_data) a = SignatureFunction(new_data)
return SignatureFunction(new_data) counter = collections.Counter()
counter.subtract(self.cnt_signature_jumps)
sf = SignatureFunction(counter=counter)
assert a == sf
return sf
# TBD short # TBD short
def __add__(self, other): def __add__(self, other):
@ -132,15 +171,21 @@ class SignatureFunction(object):
if jump_arg not in new_data.keys(): if jump_arg not in new_data.keys():
new_data[jump_arg] = self.signature_jumps[jump_arg] new_data[jump_arg] = self.signature_jumps[jump_arg]
counter = collections.Counter() counter = copy(self.cnt_signature_jumps)
counter.update(self.counter_signature_jumps) counter.update(other.cnt_signature_jumps)
counter.update(other.counter_signature_jumps)
assert collections.defaultdict(int, counter) == new_data assert collections.defaultdict(int, counter) == new_data
return SignatureFunction(counter=counter) return SignatureFunction(counter=counter)
def __eq__(self, other):
return self.cnt_signature_jumps == other.cnt_signature_jumps
def __sub__(self, other): def __sub__(self, other):
return self + other.__neg__() a = self + other.__neg__()
counter = copy(self.cnt_signature_jumps)
counter.subtract(other.cnt_signature_jumps)
sf = SignatureFunction(counter=counter)
assert a == sf
return sf
def __str__(self): def __str__(self):
return ''.join([str(jump_arg) + ": " + str(jump) + "\n" return ''.join([str(jump_arg) + ": " + str(jump) + "\n"
@ -154,13 +199,21 @@ class SignatureFunction(object):
def __call__(self, arg): def __call__(self, arg):
# Compute the value of the signature function at the point arg. # Compute the value of the signature function at the point arg.
# This requires summing all signature jumps that occur before arg. # This requires summing all signature jumps that occur before arg.
assert 0 <= arg and arg < 1 arg = mod_one(arg)
val = 0 val = 0
for jump_arg, jump in self.signature_jumps.items(): for jump_arg, jump in self.signature_jumps.items():
if jump_arg < arg: if jump_arg < arg:
val += 2 * jump val += 2 * jump
elif jump_arg == arg: elif jump_arg == arg:
val += jump val += jump
result = 0
for jump_arg, jump in self.cnt_signature_jumps.items():
if jump_arg < arg:
result += 2 * jump
elif jump_arg == arg:
result += jump
assert val == result
return val return val
@ -199,7 +252,7 @@ def search_for_large_signature_value(knot_formula=None,
k[2] > 4 * k[1] and k[2] > 4 * k[1] and
k[1] > 4 * k[0]): k[1] > 4 * k[0]):
if verbose: if verbose:
print "Ratio-condition does not hold" print("Ratio-condition does not hold")
continue continue
result = eval_cable_for_large_signature(k_vector=k, result = eval_cable_for_large_signature(k_vector=k,
knot_formula=knot_formula, knot_formula=knot_formula,
@ -220,7 +273,7 @@ def eval_cable_for_large_signature(k_vector=None,
if k_vector is None: if k_vector is None:
if q_vector is None: if q_vector is None:
# TBD docstring # TBD docstring
print "Please give a list of k (k_vector) or q values (q_vector)." print("Please give a list of k (k_vector) or q values (q_vector).")
k = k_vector k = k_vector
knot_sum = eval(knot_formula) knot_sum = eval(knot_formula)
@ -231,15 +284,17 @@ def eval_cable_for_large_signature(k_vector=None,
ksi = 1/q_4 ksi = 1/q_4
if verbose: if verbose:
print "\n\n" print("\n\n")
print 100 * "*" print(100 * "*")
print "Searching for a large signature values for the cable sum: " print("Searching for a large signature values for the cable sum: ")
print knot_description print(knot_description)
if len(knot_sum) != 4: if len(knot_sum) != 4:
print "Wrong number of cable direct summands!" print("Wrong number of cable direct summands!")
return None return None
large_sigma_for_all_v_comninations = True
good_knots = [("nic")]
# iteration over all possible character combinations # iteration over all possible character combinations
ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum] ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum]
for v_theta in it.product(*ranges_list): for v_theta in it.product(*ranges_list):
@ -254,9 +309,15 @@ def eval_cable_for_large_signature(k_vector=None,
if (theta_squers[0] - theta_squers[1] + if (theta_squers[0] - theta_squers[1] +
theta_squers[2] - theta_squers[3]) % q_4: theta_squers[2] - theta_squers[3]) % q_4:
if verbose: if verbose:
print "The condition is not satisfied: " + \ print("The condition is not satisfied: " + \
str(condition) + " != 0." str(condition) + " != 0.")
continue continue
if v_theta[0] == v_theta[1] == v_theta[2] == v_theta[3] == 0:
print("\nSkip")
continue
if v_theta[0] == v_theta[1] == v_theta[2] == v_theta[3]:
print("\nall v == a")
# T(2, q_1; 2, q_2; 2, q_4) # -T(2, q_2; 2, q_4) # # T(2, q_1; 2, q_2; 2, q_4) # -T(2, q_2; 2, q_4) #
# # T(2, q_3; 2, q_4) # -T(2, q_1; 2, q_3; 2, q_4) # # T(2, q_3; 2, q_4) # -T(2, q_1; 2, q_3; 2, q_4)
@ -266,12 +327,12 @@ def eval_cable_for_large_signature(k_vector=None,
sigma_q_2 = get_untwisted_signature_function(k_2) sigma_q_2 = get_untwisted_signature_function(k_2)
sigma_q_3 = get_untwisted_signature_function(k_3) sigma_q_3 = get_untwisted_signature_function(k_3)
a_1, a_2, a_3, a_4 = v_theta a_1, a_2, a_3, a_4 = v_theta
untwisted_part = 2 * (sigma_q_2(mod_one(ksi * a_1)) + untwisted_part = 2 * (sigma_q_2(ksi * a_1) +
sigma_q_1(mod_one(ksi * a_1 * 2)) - sigma_q_1(ksi * a_1 * 2) -
sigma_q_2(mod_one(ksi * a_2)) + sigma_q_2(ksi * a_2) +
sigma_q_3(mod_one(ksi * a_3)) - sigma_q_3(ksi * a_3) -
sigma_q_3(mod_one(ksi * a_4)) - sigma_q_3(ksi * a_4) -
sigma_q_1(mod_one(ksi * a_4 * 2))) sigma_q_1(ksi * a_4 * 2))
# "twisted" part # "twisted" part
tp = [0, 0, 0, 0] tp = [0, 0, 0, 0]
@ -281,48 +342,57 @@ def eval_cable_for_large_signature(k_vector=None,
twisted_part = tp[0] - tp[1] + tp[2] - tp[3] twisted_part = tp[0] - tp[1] + tp[2] - tp[3]
assert twisted_part == int(twisted_part) assert twisted_part == int(twisted_part)
# y = f(*v_theta)(1/2)
sigma_v = untwisted_part + twisted_part sigma_v = untwisted_part + twisted_part
if abs(sigma_v) > 5 + np.count_nonzero(v_theta): if abs(sigma_v) > 5 + np.count_nonzero(v_theta):
if config.print_calculations_for_large_signature: if config.print_calculations_for_large_signature:
print "*" * 100 print("*" * 100)
print "\n\nLarge signature value\n" print("\n\nLarge signature value\n")
print knot_description print(knot_description)
print "\nv_theta: ", print("\nv_theta: ", end="")
print v_theta print(v_theta)
print "k values: ", print("k values: ", end="")
print str(k_1) + " " + str(k_2) + " " + \ print(str(k_1) + " " + str(k_2) + " " + \
str(k_3) + " " + str(k_4) str(k_3) + " " + str(k_4))
print condition print(condition)
print "non zero value in v_theta: " + \ print("non zero value in v_theta: " + \
str(np.count_nonzero(v_theta)) str(np.count_nonzero(v_theta)))
print "sigma_v: " + str(sigma_v) print("sigma_v: " + str(sigma_v))
print "\ntwisted_part: ", print("\ntwisted_part: ", end="")
print twisted_part print(twisted_part)
print "untwisted_part: ", print("untwisted_part: ", end="")
print untwisted_part print(untwisted_part)
print "\n\nCALCULATIONS" print("\n\nCALCULATIONS")
print "*" * 100 print("*" * 100)
print_results_LT(v_theta, knot_description, print_results_LT(v_theta, knot_description,
ksi, untwisted_part, ksi, untwisted_part,
k, sigma_q_1, sigma_q_2, sigma_q_3) k, sigma_q_1, sigma_q_2, sigma_q_3)
print_results_sigma(v_theta, knot_description, tp, q_4) print_results_sigma(v_theta, knot_description, tp, q_4)
print "*" * 100 + "\n" * 5 print("*" * 100 + "\n" * 5)
else: else:
print knot_description + "\t" + str(v_theta) +\ print(knot_description + "\t" + str(v_theta) +\
"\t" + str(sigma_v) "\t" + str(sigma_v))
if config.stop_after_firts_large_signature: if config.stop_after_firts_large_signature:
break break
else: else:
if config.print_calculations_for_small_signature: if config.print_calculations_for_small_signature:
print "\n" * 5 + "*" * 100 print("\n" * 5 + "*" * 100)
print "\nSmall signature value\n" print("\nSmall signature value\n")
print knot_description print(knot_description)
print_results_LT(v_theta, knot_description, ksi, untwisted_part, print_results_LT(v_theta, knot_description, ksi, untwisted_part,
k, sigma_q_1, sigma_q_2, sigma_q_3) k, sigma_q_1, sigma_q_2, sigma_q_3)
print_results_sigma(v_theta, knot_description, tp, q_4) print_results_sigma(v_theta, knot_description, tp, q_4)
print "*" * 100 + "\n" * 5 print("*" * 100 + "\n" * 5)
else:
print(knot_description + "\t" + str(v_theta) +\
"\t" + str(sigma_v))
large_sigma_for_all_v_comninations = False
print("ojojojoj")
break
if large_sigma_for_all_v_comninations:
print("\n\n\nHura hura")
good_knots.append((knot_description, v_theta))
# else: # else:
# print "\n\tSmall signature value" # print "\n\tSmall signature value"
@ -331,18 +401,18 @@ def eval_cable_for_large_signature(k_vector=None,
# print condition # print condition
# print "non zero value in v_theta: " + str(np.count_nonzero(v_theta)) # print "non zero value in v_theta: " + str(np.count_nonzero(v_theta))
# print "signature at 1/2: " + str(y) # print "signature at 1/2: " + str(y)
return None return good_knots
def print_results_LT(v_theta, knot_description, ksi, untwisted_part, def print_results_LT(v_theta, knot_description, ksi, untwisted_part,
k, sigma_q_1, sigma_q_2, sigma_q_3): k, sigma_q_1, sigma_q_2, sigma_q_3):
a_1, a_2, a_3, a_4 = v_theta a_1, a_2, a_3, a_4 = v_theta
k_1, k_2, k_3, k_4 = [abs(i) for i in k] k_1, k_2, k_3, k_4 = [abs(i) for i in k]
print "\n\nLevine-Tristram signatures for the cable sum: " print("\n\nLevine-Tristram signatures for the cable sum: ")
print knot_description print(knot_description)
print "and characters:\n" + str(v_theta) + "," print("and characters:\n" + str(v_theta) + ",")
print "ksi = " + str(ksi) print("ksi = " + str(ksi))
print "\n\n2 * (sigma_q_2(ksi * a_1) + " + \ print("\n\n2 * (sigma_q_2(ksi * a_1) + " + \
"sigma_q_1(ksi * a_1 * 2) - " +\ "sigma_q_1(ksi * a_1 * 2) - " +\
"sigma_q_2(ksi * a_2) + " +\ "sigma_q_2(ksi * a_2) + " +\
"sigma_q_3(ksi * a_3) - " +\ "sigma_q_3(ksi * a_3) - " +\
@ -376,71 +446,71 @@ def print_results_LT(v_theta, knot_description, ksi, untwisted_part,
str(mod_one(ksi * a_4 * 2)) + \ str(mod_one(ksi * a_4 * 2)) + \
\ \
") = \n\n2 * ((" + \ ") = \n\n2 * ((" + \
str(sigma_q_2(mod_one(ksi * a_1))) + \ str(sigma_q_2(ksi * a_1)) + \
") + (" + \ ") + (" + \
str(sigma_q_1(mod_one(ksi * a_1 * 2))) + \ str(sigma_q_1(ksi * a_1 * 2)) + \
") - (" + \ ") - (" + \
str(sigma_q_2(mod_one(ksi * a_2))) + \ str(sigma_q_2(ksi * a_2)) + \
") + (" + \ ") + (" + \
str(sigma_q_3(mod_one(ksi * a_3))) + \ str(sigma_q_3(ksi * a_3)) + \
") - (" + \ ") - (" + \
str(sigma_q_3(mod_one(ksi * a_4))) + \ str(sigma_q_3(ksi * a_4)) + \
") - (" + \ ") - (" + \
str(sigma_q_1(mod_one(ksi * a_4 * 2))) + ")) = " + \ str(sigma_q_1(ksi * a_4 * 2)) + ")) = " + \
"\n\n2 * (" + \ "\n\n2 * (" + \
str(sigma_q_2(mod_one(ksi * a_1)) + str(sigma_q_2(ksi * a_1) +
sigma_q_1(mod_one(ksi * a_1 * 2)) - sigma_q_1(ksi * a_1 * 2) -
sigma_q_2(mod_one(ksi * a_2)) + sigma_q_2(ksi * a_2) +
sigma_q_3(mod_one(ksi * a_3)) - sigma_q_3(ksi * a_3) -
sigma_q_3(mod_one(ksi * a_4)) - sigma_q_3(ksi * a_4) -
sigma_q_1(mod_one(ksi * a_4 * 2))) + \ sigma_q_1(ksi * a_4 * 2)) + \
") = " + str(untwisted_part) ") = " + str(untwisted_part))
print "\nSignatures:" print("\nSignatures:")
print "\nq_1 = " + str(2 * k_1 + 1) + ": " + repr(sigma_q_1) print("\nq_1 = " + str(2 * k_1 + 1) + ": " + repr(sigma_q_1))
print "\nq_2 = " + str(2 * k_2 + 1) + ": " + repr(sigma_q_2) print("\nq_2 = " + str(2 * k_2 + 1) + ": " + repr(sigma_q_2))
print "\nq_3 = " + str(2 * k_3 + 1) + ": " + repr(sigma_q_3) print("\nq_3 = " + str(2 * k_3 + 1) + ": " + repr(sigma_q_3))
def print_results_sigma(v_theta, knot_description, tp, q_4): def print_results_sigma(v_theta, knot_description, tp, q_4):
a_1, a_2, a_3, a_4 = v_theta a_1, a_2, a_3, a_4 = v_theta
print "\n\nSigma values for the cable sum: " print("\n\nSigma values for the cable sum: ")
print knot_description print(knot_description)
print "and characters: " + str(v_theta) print("and characters: " + str(v_theta))
print "\nsigma(T_{2, q_4}, ksi_a) = " + \ print("\nsigma(T_{2, q_4}, ksi_a) = " + \
"-q + (2 * a * (q_4 - a)/q_4) " +\ "-q + (2 * a * (q_4 - a)/q_4) " +\
"= -q + 2 * a - 2 * a^2/q_4 if a != 0,\n\t\t\t" +\ "= -q + 2 * a - 2 * a^2/q_4 if a != 0,\n\t\t\t" +\
" = 0 if a == 0." " = 0 if a == 0.")
print "\nsigma(T_{2, q_4}, chi_a_1) = ", print("\nsigma(T_{2, q_4}, chi_a_1) = ", end="")
if a_1: if a_1:
print "- (" + str(q_4) + ") + 2 * " + str(a_1) + " + " +\ print("- (" + str(q_4) + ") + 2 * " + str(a_1) + " + " +\
"- 2 * " + str(a_1^2) + "/" + str(q_4) + \ "- 2 * " + str(a_1^2) + "/" + str(q_4) + \
" = " + str(tp[0]) " = " + str(tp[0]))
else: else:
print "0" print("0")
print "\nsigma(T_{2, q_4}, chi_a_2) = ", print("\nsigma(T_{2, q_4}, chi_a_2) = ", end ="")
if a_2: if a_2:
print "- (" + str(q_4) + ") + 2 * " + str(a_2) + " + " +\ print("- (" + str(q_4) + ") + 2 * " + str(a_2) + " + " +\
"- 2 * " + str(a_2^2) + "/" + str(q_4) + \ "- 2 * " + str(a_2^2) + "/" + str(q_4) + \
" = " + str(tp[1]) " = " + str(tp[1]))
else: else:
print "0", print("0", end="")
print "\nsigma(T_{2, q_4}, chi_a_3) = ", print("\nsigma(T_{2, q_4}, chi_a_3) = ", end="")
if a_3: if a_3:
print "- (" + str(q_4) + ") + 2 * " + str(a_3) + " + " +\ print("- (" + str(q_4) + ") + 2 * " + str(a_3) + " + " +\
"- 2 * " + str(a_3^2) + "/" + str(q_4) + \ "- 2 * " + str(a_3^2) + "/" + str(q_4) + \
" = " + str(tp[2]) " = " + str(tp[2]))
else: else:
print "0", print("0", end="")
print "\nsigma(T_{2, q_4}, chi_a_4) = ", print("\nsigma(T_{2, q_4}, chi_a_4) = ", end="")
if a_4: if a_4:
print "- (" + str(q_4) + ") + 2 * " + str(a_4) + " + " +\ print("- (" + str(q_4) + ") + 2 * " + str(a_4) + " + " +\
"- 2 * " + str(a_4^2) + "/" + str(q_4) + \ "- 2 * " + str(a_4^2) + "/" + str(q_4) + \
" = " + str(tp[3]) " = " + str(tp[3]))
else: else:
print "0" print("0")
print "\n\nsigma(T_{2, q_4}, chi_a_1) " + \ print("\n\nsigma(T_{2, q_4}, chi_a_1) " + \
"- sigma(T_{2, q_4}, chi_a_2) " + \ "- sigma(T_{2, q_4}, chi_a_2) " + \
"+ sigma(T_{2, q_4}, chi_a_3) " + \ "+ sigma(T_{2, q_4}, chi_a_3) " + \
"- sigma(T_{2, q_4}, chi_a_4) =\n" + \ "- sigma(T_{2, q_4}, chi_a_4) =\n" + \
@ -448,7 +518,7 @@ def print_results_sigma(v_theta, knot_description, tp, q_4):
") - sigma(T_{2, q_4}, " + str(a_2) + \ ") - sigma(T_{2, q_4}, " + str(a_2) + \
") + sigma(T_{2, q_4}, " + str(a_3) + \ ") + sigma(T_{2, q_4}, " + str(a_3) + \
") - sigma(T_{2, q_4}, " + str(a_4) + ") = " + \ ") - sigma(T_{2, q_4}, " + str(a_4) + ") = " + \
str(tp[0] - tp[1] + tp[2] - tp[3]) str(tp[0] - tp[1] + tp[2] - tp[3]))
# searching for signature == 0 # searching for signature == 0
def search_for_null_signature_value(knot_formula=None, limit=None): def search_for_null_signature_value(knot_formula=None, limit=None):
@ -468,7 +538,7 @@ def search_for_null_signature_value(knot_formula=None, limit=None):
k = get_shifted_combination(k) k = get_shifted_combination(k)
knot_sum = eval(knot_formula) knot_sum = eval(knot_formula)
if is_trivial_combination(knot_sum): if is_trivial_combination(knot_sum):
print knot_sum print(knot_sum)
continue continue
result = eval_cable_for_null_signature(knot_sum) result = eval_cable_for_null_signature(knot_sum)
@ -492,8 +562,8 @@ def eval_cable_for_null_signature(knot_sum, print_results=False, verbose=None):
ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum] ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum]
if verbose: if verbose:
print print()
print knot_description print(knot_description)
for v_theta in it.product(*ranges_list): for v_theta in it.product(*ranges_list):
if f(*v_theta, verbose=False).is_zero_everywhere(): if f(*v_theta, verbose=False).is_zero_everywhere():
zero_theta_combinations.append(v_theta) zero_theta_combinations.append(v_theta)
@ -503,14 +573,14 @@ def eval_cable_for_null_signature(knot_sum, print_results=False, verbose=None):
# assert sum(v_theta) != 0 # assert sum(v_theta) != 0
if print_results: if print_results:
print print()
print knot_description print(knot_description)
print "Zero cases: " + str(null_combinations) print("Zero cases: " + str(null_combinations))
print "All cases: " + str(all_combinations) print("All cases: " + str(all_combinations))
if zero_theta_combinations: if zero_theta_combinations:
print "Zero theta combinations: " print("Zero theta combinations: ")
for el in zero_theta_combinations: for el in zero_theta_combinations:
print el print(el)
if null_combinations^2 >= all_combinations: if null_combinations^2 >= all_combinations:
return knot_description, null_combinations, all_combinations return knot_description, null_combinations, all_combinations
return None return None
@ -553,12 +623,12 @@ def get_blanchfield_for_pattern(k_n, theta):
results.append((1 - e * ksi, -1 * sgn(k_n))) results.append((1 - e * ksi, -1 * sgn(k_n)))
# lambda_even # lambda_even
# print "normal" # print("normal")
for e in range(1, theta): for e in range(1, theta):
if (theta + e) % 2 == 0: if (theta + e) % 2 == 0:
results.append((e * ksi, 1 * sgn(k_n))) results.append((e * ksi, 1 * sgn(k_n)))
results.append((1 - e * ksi, -1 * sgn(k_n))) results.append((1 - e * ksi, -1 * sgn(k_n)))
# print "reversed" # print("reversed")
for e in range(theta + 1, k + 1): for e in range(theta + 1, k + 1):
if (theta + e) % 2 != 0: if (theta + e) % 2 != 0:
continue continue
@ -637,13 +707,13 @@ def get_signature_as_theta_function(*arg, **key_args):
sf += get_signature_summand_as_theta_function(*knot)(thetas[i]) sf += get_signature_summand_as_theta_function(*knot)(thetas[i])
# in case wrong theata value was given # in case wrong theata value was given
except ValueError as e: except ValueError as e:
print "ValueError: " + str(e.args[0]) +\ print("ValueError: " + str(e.args[0]) +\
" Please change " + str(i + 1) + ". parameter." " Please change " + str(i + 1) + ". parameter.")
return None return None
if verbose: if verbose:
print print()
print str(thetas) print(str(thetas))
print sf print(sf)
return sf return sf
signature_as_theta_function.__doc__ = signature_as_theta_function_docstring signature_as_theta_function.__doc__ = signature_as_theta_function_docstring
return signature_as_theta_function return signature_as_theta_function
@ -799,7 +869,7 @@ get_signature_as_theta_function.__doc__ = \
sage: signature_function_generator = get_signature_as_theta_function( sage: signature_function_generator = get_signature_as_theta_function(
[1, 3], [2], [-1, -2], [-3]) [1, 3], [2], [-1, -2], [-3])
sage: sf = signature_function_generator(2, 1, 2, 2) sage: sf = signature_function_generator(2, 1, 2, 2)
sage: print sf sage: print(sf)
0: 0 0: 0
5/42: 1 5/42: 1
1/7: 0 1/7: 0
@ -819,8 +889,8 @@ get_signature_as_theta_function.__doc__ = \
37/42: -1 37/42: -1
Or like below. Or like below.
sage: print get_signature_as_theta_function([1, 3], [2], [-1, -2], [-3] sage: print(get_signature_as_theta_function([1, 3], [2], [-1, -2], [-3]
)(2, 1, 2, 2) )(2, 1, 2, 2))
0: 0 0: 0
1/7: 0 1/7: 0
1/6: 0 1/6: 0