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end of implementation for large gignature

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This commit is contained in:
Maria Marchwicka 2020-09-07 17:02:48 +02:00
parent 1ec26c1b55
commit ac1a1d3475
2 changed files with 256 additions and 161 deletions
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cable_signature.sage
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@ -27,6 +27,17 @@ class TorusCable(object):
self.__sigma_function = None self.__sigma_function = None
self.__signature_as_function_of_theta = None self.__signature_as_function_of_theta = None
def get_untwisted_signature_function(self, j, q=None):
# return the signature function of the T_{2,2k+1} torus knot
k = abs(j)
q = 2 * k + 1
w = ([((2 * a + 1)/(2 * q), -1 * sgn(j)) for a in range(k)] +
[((2 * a + 1)/(2 * q), 1 * sgn(j))
for a in range(k + 1, 2 * k + 1)])
return SignatureFunction(values=w)
def get_knot_descrption(self): def get_knot_descrption(self):
description = "" description = ""
for knot in self.knot_sum: for knot in self.knot_sum:
@ -63,17 +74,20 @@ class TorusCable(object):
return signature_as_function_of_theta(*(len_a * [0])) return signature_as_function_of_theta(*(len_a * [0]))
if len_t != len_a: if len_t != len_a:
msg = "This function takes exactly " + str(len_a) + \ if len(thetas[0]) == len_a:
" arguments or no argument at all (" + str(len_t) + \ thetas = thetas[0]
" given)." else:
raise TypeError(msg) msg = "This function takes exactly " + str(len_a) + \
" arguments or no argument at all (" + str(len_t) + \
" given)."
raise TypeError(msg)
sf = SignatureFunction() sf = SignatureFunction()
# for each cable knot in cable sum apply theta # for each cable knot in cable sum apply theta
for i, knot in enumerate(self.knot_sum): for i, knot in enumerate(self.knot_sum):
try: try:
ssf = get_signature_summand_as_theta_function(*knot) ssf = get_summand_signature_as_theta_function(*knot)
sf += ssf(thetas[i]) sf += ssf(thetas[i])
# in case wrong theata value was given # in case wrong theata value was given
except ValueError as e: except ValueError as e:
@ -102,6 +116,94 @@ class TorusCable(object):
number_of_null_comb += 2^m number_of_null_comb += 2^m
return number_of_null_comb, list_of_good_vectors return number_of_null_comb, list_of_good_vectors
def eval_cable_for_large_signature(self, list_of_ranges,
print_results=False,
verbose=False):
if self.__signature_as_function_of_theta is None:
self.__signature_as_function_of_theta= \
self.__get_signature_as_function_of_theta()
if print_results:
print()
print(self.knot_description, end="\t\t\t")
print()
f = self.__signature_as_function_of_theta
if self.s__check_all_combinations_in_ranges(list_of_ranges,
print_results=print_results):
return True
return False
def s__check_all_combinations_in_ranges(self, list_of_ranges,
print_results=False):
all_combinations_pass = True
all_bad_vectors = []
number_of_all_good_v = 0
for i, range_product in enumerate(list_of_ranges):
good_v, bad_v = self.s__check_combinations_in_range(range_product)
number_of_all_good_v += len(good_v)
all_bad_vectors = list(it.chain(all_bad_vectors, bad_v))
if bad_v:
all_combinations_pass = False
# if print_results:
# print("good : bad:\t " + str(len(good_v)) +\
# " : " + str(len(bad_v)))
# if i in [0, 4,]:
# print()
# if bad_v:
# print(bad_v)
if print_results:
print("good : bad:\t " + str(number_of_all_good_v) +\
" : " + str(len(all_bad_vectors)))
return all_combinations_pass
def s__check_combinations_in_range(self, range_product):
large_sigma_for_all_combinations = True
bad_vectors = []
good_vectors = []
q_4 = self.q_vector[-1]
for vector in range_product:
a_1, a_2, a_3, a_4 = vector
if (a_1^2 - a_2^2 + a_3^2 - a_4^2) % q_4:
continue
if all(a in [1, q_4 - 1] for a in vector):
pass
else:
continue
if self.s__is_sigma_for_vector_class_big(vector):
good_vectors.append(vector)
else:
# print(vector)
bad_vectors.append(vector)
return good_vectors, bad_vectors
def s__is_sigma_for_vector_class_big(self, theta_vector):
[a_1, a_2, a_3, a_4] = theta_vector
q_4 = self.q_vector[-1]
k_4 = self.k_vector[-1]
max_sigma = 0
print(theta_vector, end="\t")
for shift in range(1, k_4 + 1):
shifted_theta = [(shift * a) % q_4 for a in
[a_1, a_2, a_3, a_4]]
sf = self.__signature_as_function_of_theta(shifted_theta)
sigma_v = sf.is_big()
print(sigma_v, end=" ")
if abs(sigma_v) > abs(max_sigma):
max_sigma = sigma_v
if abs(sigma_v) > 5 + np.count_nonzero(shifted_theta):
print("\tok " + str(sigma_v))
return True
print("\tbad class " + str(max_sigma))
return False
# searching for signature == 0 # searching for signature == 0
def eval_cable_for_null_signature(self, print_results=False, verbose=False): def eval_cable_for_null_signature(self, print_results=False, verbose=False):
# search for zero combinations # search for zero combinations
@ -114,9 +216,10 @@ class TorusCable(object):
print(self.knot_description) print(self.knot_description)
print("Zero cases: " + str(number_of_null_comb)) print("Zero cases: " + str(number_of_null_comb))
print("All cases: " + str(number_of_all_comb)) print("All cases: " + str(number_of_all_comb))
print("Zero theta combinations: ") if list_of_good_vectors:
for el in list_of_good_vectors: print("Zero theta combinations: ")
print(el) for el in list_of_good_vectors:
print(el)
if number_of_null_comb^2 >= number_of_all_comb: if number_of_null_comb^2 >= number_of_all_comb:
return number_of_null_comb, number_of_all_comb return number_of_null_comb, number_of_all_comb
return None return None
@ -124,8 +227,9 @@ class TorusCable(object):
# check sigma for all v = s * [a_1, a_2, a_3, a_4] for s in [1, q_4 - 1] # check sigma for all v = s * [a_1, a_2, a_3, a_4] for s in [1, q_4 - 1]
def __is_sigma_for_vector_class_big(self, theta_vector): def __is_sigma_for_vector_class_big(self, theta_vector):
[a_1, a_2, a_3, a_4] = theta_vector [a_1, a_2, a_3, a_4] = theta_vector
q_4 = self.q_vector[3] q_4 = self.q_vector[-1]
for shift in range(1, q_4): k_4 = self.k_vector[-1]
for shift in range(1, k_4 + 1):
shifted_theta = [(shift * a) % q_4 for a in shifted_theta = [(shift * a) % q_4 for a in
[a_1, a_2, a_3, a_4]] [a_1, a_2, a_3, a_4]]
sigma_v = self.__sigma_function(shifted_theta) sigma_v = self.__sigma_function(shifted_theta)
@ -147,9 +251,6 @@ class TorusCable(object):
print("\n") print("\n")
def __tmp_get_max_sigma_for_vector_class(self, theta_vector): def __tmp_get_max_sigma_for_vector_class(self, theta_vector):
# print("\n")
# print(self.knot_description)
# print("vector = " + str(theta_vector))
max_sigma = (theta_vector, 0) max_sigma = (theta_vector, 0)
[a_1, a_2, a_3, a_4] = theta_vector [a_1, a_2, a_3, a_4] = theta_vector
q_4 = self.q_vector[3] q_4 = self.q_vector[3]
@ -159,13 +260,8 @@ class TorusCable(object):
sigma = self.__sigma_function(shifted_theta) sigma = self.__sigma_function(shifted_theta)
if abs(sigma) > abs(max_sigma[1]): if abs(sigma) > abs(max_sigma[1]):
max_sigma = (shifted_theta, sigma) max_sigma = (shifted_theta, sigma)
assert max_sigma[1] == 0, knot_description
# print("\n" + self.knot_description + "\t" + str(max_sigma[0]) +\
# "\t" + str(max_sigma[1]))
return max_sigma[1] return max_sigma[1]
def is_sigma_for_vector_class_big(self, theta_vector): def is_sigma_for_vector_class_big(self, theta_vector):
if self.__sigma_function is None: if self.__sigma_function is None:
self.__sigma_function = self.__get_sigma_function() self.__sigma_function = self.__get_sigma_function()
@ -175,9 +271,9 @@ class TorusCable(object):
k_1, k_2, k_3, k_4 = [abs(k) for k in self.k_vector] k_1, k_2, k_3, k_4 = [abs(k) for k in self.k_vector]
q_4 = 2 * k_4 + 1 q_4 = 2 * k_4 + 1
ksi = 1/q_4 ksi = 1/q_4
sigma_q_1 = get_untwisted_signature_function(k_1) sigma_q_1 = self.get_untwisted_signature_function(k_1)
sigma_q_2 = get_untwisted_signature_function(k_2) sigma_q_2 = self.get_untwisted_signature_function(k_2)
sigma_q_3 = get_untwisted_signature_function(k_3) sigma_q_3 = self.get_untwisted_signature_function(k_3)
def sigma_function(theta_vector, print_results=False): def sigma_function(theta_vector, print_results=False):
# "untwisted" part (Levine-Tristram signatures) # "untwisted" part (Levine-Tristram signatures)
@ -208,9 +304,9 @@ class TorusCable(object):
a_1, a_2, a_3, a_4 = theta_vector a_1, a_2, a_3, a_4 = theta_vector
q_4 = 2 * k_4 + 1 q_4 = 2 * k_4 + 1
ksi = 1/q_4 ksi = 1/q_4
sigma_q_1 = get_untwisted_signature_function(k_1) sigma_q_1 = self.get_untwisted_signature_function(k_1)
sigma_q_2 = get_untwisted_signature_function(k_2) sigma_q_2 = self.get_untwisted_signature_function(k_2)
sigma_q_3 = get_untwisted_signature_function(k_3) sigma_q_3 = self.get_untwisted_signature_function(k_3)
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(theta_vector) + ",") print("and characters:\n" + str(theta_vector) + ",")
@ -335,10 +431,8 @@ class TorusCable(object):
self.__sigma_function = self.__get_sigma_function() self.__sigma_function = self.__get_sigma_function()
return self.__sigma_function(theta_vector) return self.__sigma_function(theta_vector)
# searching for sigma > 5 + #(v_i != 0) # searching for sigma > 5 + #(v_i != 0)
def __check_combinations_in_range(self, range_product): def __check_combinations_in_range(self, range_product):
large_sigma_for_all_combinations = True
bad_vectors = [] bad_vectors = []
good_vectors = [] good_vectors = []
q_4 = self.q_vector[-1] q_4 = self.q_vector[-1]
@ -346,19 +440,12 @@ class TorusCable(object):
a_1, a_2, a_3, a_4 = vector a_1, a_2, a_3, a_4 = vector
if (a_1^2 - a_2^2 + a_3^2 - a_4^2) % q_4: if (a_1^2 - a_2^2 + a_3^2 - a_4^2) % q_4:
continue continue
if all(a in [1, q_4 - 1] for a in vector): # if all(a in [1, q_4 - 1] for a in vector):
continue # continue
if self.__is_sigma_for_vector_class_big(vector): if self.__is_sigma_for_vector_class_big(vector):
good_vectors.append(vector) good_vectors.append(vector)
# pass
else: else:
bad_vectors.append(vector) bad_vectors.append(vector)
# large_sigma_for_all_combinations = False
# if len(bad_vectors) > 8:
# break
print(len(bad_vectors))
if a_1 and a_2 and a_3 and a_4:
assert len(bad_vectors) % 8 == 0
return good_vectors, bad_vectors return good_vectors, bad_vectors
# searching for sigma > 5 + #(v_i != 0) # searching for sigma > 5 + #(v_i != 0)
@ -369,7 +456,7 @@ class TorusCable(object):
# searching for sigma > 5 + #(v_i != 0) # searching for sigma > 5 + #(v_i != 0)
def __check_all_combinations_in_ranges(self, list_of_ranges, def __check_all_combinations_in_ranges(self, list_of_ranges,
print_results=True): print_results=False):
all_combinations_pass = True all_combinations_pass = True
all_bad_vectors = [] all_bad_vectors = []
number_of_all_good_v = 0 number_of_all_good_v = 0
@ -379,8 +466,6 @@ class TorusCable(object):
all_bad_vectors = list(it.chain(all_bad_vectors, bad_v)) all_bad_vectors = list(it.chain(all_bad_vectors, bad_v))
if bad_v: if bad_v:
all_combinations_pass = False all_combinations_pass = False
if len(all_bad_vectors) > 8:
break
# if print_results: # if print_results:
# print("good : bad:\t " + str(len(good_v)) +\ # print("good : bad:\t " + str(len(good_v)) +\
# " : " + str(len(bad_v))) # " : " + str(len(bad_v)))
@ -392,11 +477,6 @@ class TorusCable(object):
if print_results: if print_results:
print("good : bad:\t " + str(number_of_all_good_v) +\ print("good : bad:\t " + str(number_of_all_good_v) +\
" : " + str(len(all_bad_vectors))) " : " + str(len(all_bad_vectors)))
# if len(all_bad_vectors) < 8:
# print()
# print(all_bad_vectors)
return all_combinations_pass return all_combinations_pass
# searching for sigma > 5 + #(v_i != 0) # searching for sigma > 5 + #(v_i != 0)
@ -405,11 +485,7 @@ class TorusCable(object):
if self.__sigma_function is None: if self.__sigma_function is None:
self.__sigma_function = self.__get_sigma_function() self.__sigma_function = self.__get_sigma_function()
if print_results: if print_results:
# print("\n\n")
# print(100 * "*")
# print("Searching for a large signature values for the cable sum: ")
print(self.knot_description, end="\t\t\t") print(self.knot_description, end="\t\t\t")
# print()
if self.__check_all_combinations_in_ranges(list_of_ranges, if self.__check_all_combinations_in_ranges(list_of_ranges,
print_results=print_results): print_results=print_results):
return True return True
@ -459,16 +535,16 @@ class SignatureFunction(object):
counter[mod_one(2 * jump_arg)] = jump counter[mod_one(2 * jump_arg)] = jump
return SignatureFunction(counter=counter) return SignatureFunction(counter=counter)
def __lshift__(self, shift):
# A shift of the signature functions corresponds to the rotation.
return self.__rshift__(-shift)
def __rshift__(self, shift): def __rshift__(self, shift):
# A shift of the signature functions corresponds to the rotation.
new_data = [] new_data = []
for jump_arg, jump in self.cnt_signature_jumps.items(): 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(values=new_data) return SignatureFunction(values=new_data)
def __lshift__(self, shift):
return self.__rshift__(-shift)
def __neg__(self): def __neg__(self):
counter = collections.Counter() counter = collections.Counter()
counter.subtract(self.cnt_signature_jumps) counter.subtract(self.cnt_signature_jumps)
@ -498,16 +574,33 @@ class SignatureFunction(object):
return result[:-2] + "." return result[:-2] + "."
def __call__(self, arg): def __call__(self, arg):
# Compute the value of the signature function at the point arg. # return the value of the signature function at the point arg, i.e.
# This requires summing all signature jumps that occur before arg. # sum of all signature jumps that occur before arg
arg = mod_one(arg) arg = mod_one(arg)
cnt = self.cnt_signature_jumps cnt = self.cnt_signature_jumps
before_arg = [jump for jump_arg, jump in cnt.items() if jump_arg < arg] before_arg = [jump for jump_arg, jump in cnt.items() if jump_arg < arg]
return 2 * sum(before_arg) + cnt[arg] return 2 * sum(before_arg) + cnt[arg]
def is_big(self):
max = 0
items = self.cnt_signature_jumps.items()
for arg, _ in items:
current = sum([jump for jump_arg, jump in items if jump_arg <= arg])
if abs(current) > abs(max):
max = current
if abs(max) > 9:
return max
return max
def mod_one(n): def mod_one(n):
return n - floor(n) return n - floor(n)
def get_untwisted_signature_function(j): def get_untwisted_signature_function(j):
# return the signature function of the T_{2,2k+1} torus knot # return the signature function of the T_{2,2k+1} torus knot
k = abs(j) k = abs(j)
@ -517,15 +610,16 @@ def get_untwisted_signature_function(j):
return SignatureFunction(values=w) return SignatureFunction(values=w)
def get_signature_summand_as_theta_function(*arg): def get_summand_signature_as_theta_function(*arg):
def get_signture_function(theta): def get_summand_signture_function(theta):
# TBD: another formula (for t^2) description # TBD: another formula (for t^2) description
k_n = abs(arg[-1]) k_n = abs(arg[-1])
if theta > k_n: if theta > k_n:
msg = "k for the pattern in the cable is " + str(arg[-1]) + \ msg = "k for the pattern in the cable is " + str(arg[-1]) + \
". Parameter theta should not be larger than abs(k)." ". Parameter theta should not be larger than abs(k)."
raise ValueError(msg) pass
# print(msg)
# raise ValueError(msg)
# twisted part # twisted part
cable_signature = get_blanchfield_for_pattern(arg[-1], theta) cable_signature = get_blanchfield_for_pattern(arg[-1], theta)
@ -546,8 +640,8 @@ def get_signature_summand_as_theta_function(*arg):
test2 = -c + b test2 = -c + b
assert test == test assert test == test
return cable_signature return cable_signature
get_signture_function.__doc__ = get_signture_function_docsting get_summand_signture_function.__doc__ = get_summand_signture_function_docsting
return get_signture_function return get_summand_signture_function
def get_blanchfield_for_pattern(k_n, theta): def get_blanchfield_for_pattern(k_n, theta):
@ -580,15 +674,17 @@ def get_blanchfield_for_pattern(k_n, theta):
return SignatureFunction(values=results) return SignatureFunction(values=results)
def get_signature_summand_as_theta_function(*arg): def get_summand_signature_as_theta_function(*arg):
def get_signture_function(theta): def get_summand_signture_function(theta):
# TBD: another formula (for t^2) description # TBD: another formula (for t^2) description
k_n = abs(arg[-1]) k_n = abs(arg[-1])
if theta > k_n: if theta > k_n:
msg = "k for the pattern in the cable is " + str(arg[-1]) + \ msg = "k for the pattern in the cable is " + str(arg[-1]) + \
". Parameter theta should not be larger than abs(k)." ". Parameter theta should not be larger than abs(k)."
raise ValueError(msg) pass
# print(msg)
# raise ValueError(msg)
# twisted part # twisted part
cable_signature = get_blanchfield_for_pattern(arg[-1], theta) cable_signature = get_blanchfield_for_pattern(arg[-1], theta)
@ -609,8 +705,8 @@ def get_signature_summand_as_theta_function(*arg):
test2 = -c + b test2 = -c + b
assert test == test assert test == test
return cable_signature return cable_signature
get_signture_function.__doc__ = get_signture_function_docsting get_summand_signture_function.__doc__ = get_summand_signture_function_docsting
return get_signture_function return get_summand_signture_function
def get_untwisted_signature_function(j): def get_untwisted_signature_function(j):
@ -733,12 +829,12 @@ SignatureFunction.__doc__ = \
signature functions as defined on the interval [0,1). signature functions as defined on the interval [0,1).
""" """
get_signture_function_docsting = \ get_summand_signture_function_docsting = \
""" """
This function returns SignatureFunction for previously defined single This function returns SignatureFunction for previously defined single
cable T_(2, q) and a theta given as an argument. cable T_(2, q) and a theta given as an argument.
The cable was defined by calling function The cable was defined by calling function
get_signature_summand_as_theta_function(*arg) get_summand_signature_as_theta_function(*arg)
with the cable description as an argument. with the cable description as an argument.
It is an implementaion of the formula: It is an implementaion of the formula:
Bl_theta(K'_(2, d)) = Bl_theta(K'_(2, d)) =
@ -785,7 +881,7 @@ get_blanchfield_for_pattern.__doc__ = \
(https://arxiv.org/pdf/1809.08791.pdf) (https://arxiv.org/pdf/1809.08791.pdf)
""" """
get_signature_summand_as_theta_function.__doc__ = \ get_summand_signature_as_theta_function.__doc__ = \
""" """
Argument: Argument:
n integers that encode a single cable, i.e. n integers that encode a single cable, i.e.

187
my_signature.sage
View File

@ -1,7 +1,6 @@
#!/usr/bin/python #!/usr/bin/python
# TBD: read about Factory Method, variable in docstring, sage documentation # TBD: read about Factory Method, variable in docstring, sage documentation
# move settings to sep file
import os import os
import sys import sys
@ -13,7 +12,7 @@ import re
# os.system('sage --preparse cable_signature.sage') # os.system('sage --preparse cable_signature.sage')
# os.system('mv cable_signature.sage.py cable_signature.py') # os.system('mv cable_signature.sage.py cable_signature.py')
# from cable_signature import SignatureFunction, TorusCable # from cable_signature import SignatureFunction, TorusCable
#
class Config(object): class Config(object):
def __init__(self): def __init__(self):
@ -42,115 +41,58 @@ class Config(object):
self.verbose = False self.verbose = False
self.print_results = True self.print_results = True
self.print_results = False # self.print_results = False
self.print_calculations_for_large_sigma = True self.print_calculations_for_large_sigma = True
self.print_calculations_for_large_sigma = False self.print_calculations_for_large_sigma = False
# is the ratio restriction for values in k_vector taken into account # is the ratio restriction for values in q_vector taken into account
# 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
# range for a_i, v = [a_1, a_2, a_3, a_4], for sigma calculations # range for a_i, v = [a_1, a_2, a_3, a_4], for sigma calculations
def get_list_of_ranges(self, q): # upper bound supposed to be ub = k + 1
def get_list_of_ranges(self, ub):
list_of_ranges = [ list_of_ranges = [
# all characters a_1, a_2, a_3, a_4 != 0 # all characters a_1, a_2, a_3, a_4 != 0
it.product(range(1, q), range(1, q), range(1, q), range(1, 2)), it.product(range(1, ub), range(1, ub), range(1, ub), range(1, 2)),
# a_1 == 0, a_2, a_3, a_4 != 0 # a_1 == 0, a_2, a_3, a_4 != 0
it.product(range(1), range(1, q), range(1, q), range(1, 2)), it.product(range(1), range(1, ub), range(1, ub), range(1, 2)),
# a_2 == 0, a_1, a_3, a_4 != 0 # a_2 == 0, a_1, a_3, a_4 != 0
it.product(range(1, q), range(1), range(1, q), range(1, 2)), it.product(range(1, ub), range(1), range(1, ub), range(1, 2)),
# a_3 == 0, a_1, a_2, a_4 != 0 # a_3 == 0, a_1, a_2, a_4 != 0
it.product(range(1, q), range(1, q), range(1), range(1, 2)), it.product(range(1, ub), range(1, ub), range(1), range(1, 2)),
# a_4 == 0, a_1, a_2, a_3 != 0 # a_4 == 0, a_1, a_2, a_3 != 0
it.product(range(1, q), range(1, q), range(1, 2), range(1)), it.product(range(1, ub), range(1, ub), range(1, 2), range(1)),
# a_1 == 0, a_2 == 0, a_3, a_4 != 0 # a_1 == 0, a_2 == 0, a_3, a_4 != 0
it.product(range(1), range(1), range(1, q), range(1, 2)), it.product(range(1), range(1), range(1, ub), range(1, 2)),
# a_1 == 0, a_3 == 0, a_2, a_4 != 0 # a_1 == 0, a_3 == 0, a_2, a_4 != 0
it.product(range(1), range(1, q), range(1), range(1, 2)), it.product(range(1), range(1, ub), range(1), range(1, 2)),
# a_1 == 0, a_4 == 0, a_3, a_2 != 0 # a_1 == 0, a_4 == 0, a_3, a_2 != 0
it.product(range(1), range(1, q), range(1, 2), range(1)), it.product(range(1), range(1, ub), range(1, 2), range(1)),
# a_2 == 0, a_3 == 0, a_1, a_4 != 0 # a_2 == 0, a_3 == 0, a_1, a_4 != 0
it.product(range(1, q), range(1), range(1), range(1, 2)), it.product(range(1, ub), range(1), range(1), range(1, 2)),
# a_2 == 0, a_4 == 0, a_1, a_3 != 0 # a_2 == 0, a_4 == 0, a_1, a_3 != 0
it.product(range(1, q), range(1), range(1, 2), range(1)), it.product(range(1, ub), range(1), range(1, 2), range(1)),
# a_3 == 0, a_4 == 0, a_1, a_2 != 0 # a_3 == 0, a_4 == 0, a_1, a_2 != 0
it.product(range(1, q), range(1, 2), range(1), range(1)), it.product(range(1, ub), range(1, 2), range(1), range(1)),
] ]
list_of_ranges =
[
# all characters a_1, a_2, a_3, a_4 != 0
# 1, 1, 1, 1
it.product(range(1, 2), range(1, 2), range(1, 2), range(1, 2)),
# -1, -1, -1, 1
it.product(range(q - 1, q), range(q - 1, q), range(q - 1, q), \
range(1, 2)),
# 1, -1, -1, 1
it.product(range(1, 2), range(q - 1, q), range(q - 1, q), \
range(1, 2)),
# -1 , -1, 1, 1
it.product(range(q - 1, q), range(q - 1, q), range(1, 2), \
range(1, 2)),
# -1, 1, -1, 1
it.product(range(q - 1, q), range(1, 2), range(q - 1, q), \
range(1, 2)),
# 1, 1, -1, 1
it.product(range(1, 2), range(1, 2), range(q - 1, q), \
range(1, 2)),
# 1, -1, 1, 1
it.product(range(1, 2), range(q - 1, q), range(1, 2), \
range(1, 2)),
# -1, 1, 1, 1
it.product(range(q - 1, q), range(1, 2), range(1, 2), \
range(1, 2)),
]
return list_of_ranges return list_of_ranges
def main(arg): def main(arg):
if arg[1]: try:
limit = int(arg[1]) limit = int(arg[1])
else: except IndexError:
limit = None limit = None
knots_with_large_sigma = search_for_large_signature_value(limit=limit) search_for_large_signature_value(limit=limit)
knots_with_large_sigma = search_for_large_sigma_value(limit=limit)
# search_for_null_signature_value(limit=limit) # search_for_null_signature_value(limit=limit)
# searching for sigma > 5 + #(v_i != 0) over given knot schema def set_parameters(knot_formula, limit, verbose, print_results):
def __search_for_large_signature_value(knot_formula, limit,
verbose, print_results):
# number of k_i (q_i) variables to substitute
k_size = extract_max(knot_formula) + 1
combinations = it.combinations_with_replacement(range(0, limit + 1), k_size)
P = Primes()
good_knots = []
# iterate over q-vector
for c in combinations:
q = list(c)
q[0] = P.unrank(q[0] + 1 + config.start_shift)
q[1] = P.next(q[0] * 4 + q[1])
q[2] = P.next(q[1] * 4 + q[2])
q[3] = P.next(q[2] * 4 + q[3])
cable = TorusCable(knot_formula=knot_formula, q_vector=q)
list_of_ranges = config.get_list_of_ranges(cable.q_vector[-1])
print(cable.knot_description)
if cable.eval_cable_for_large_sigma(list_of_ranges, verbose=verbose,
print_results=print_results):
good_knots.append(cable)
return good_knots
# searching for sigma > 5 + #(v_i != 0) over given knot schema
def search_for_large_signature_value(knot_formula=None, limit=None,
verbose=None, print_results=None):
if limit is None: if limit is None:
limit = config.limit limit = config.limit
if knot_formula is None: if knot_formula is None:
@ -159,12 +101,19 @@ def search_for_large_signature_value(knot_formula=None, limit=None,
vebose = config.verbose vebose = config.verbose
if print_results is None: if print_results is None:
print_results = config.print_results print_results = config.print_results
return knot_formula, limit, verbose, print_results
# searching for sigma > 5 + #(v_i != 0) over given knot schema
def search_for_large_sigma_value(knot_formula=None, limit=None,
verbose=None, print_results=None):
knot_formula, limit, verbose, print_results = \
set_parameters(knot_formula, limit, verbose, print_results)
k_vector_size = extract_max(knot_formula) + 1 k_vector_size = extract_max(knot_formula) + 1
limit = max(limit, k_vector_size) limit = max(limit, k_vector_size)
if config.only_slice_candidates:
return __search_for_large_signature_value(knot_formula, limit, verbose,
print_results)
# number of k_i (q_i) variables to substitute # number of k_i (q_i) variables to substitute
combinations = it.combinations(range(1, limit + 1), k_vector_size) combinations = it.combinations(range(1, limit + 1), k_vector_size)
@ -173,23 +122,27 @@ def search_for_large_signature_value(knot_formula=None, limit=None,
# iterate over q-vector # iterate over q-vector
for c in combinations: for c in combinations:
k = [(P.unrank(i + config.start_shift) - 1)/2 for i in c] q = [P.unrank(i + config.start_shift) for i in c]
cable = TorusCable(knot_formula=knot_formula, k_vector=k) if config.only_slice_candidates:
list_of_ranges = config.get_list_of_ranges(cable.q_vector[-1]) if not (q[3] > 4 * q[2] and
print(cable.knot_description) q[2] > 4 * q[1] and
q[1] > 4 * q[0]):
if verbose:
print("Ratio-condition does not hold")
continue
cable = TorusCable(knot_formula=knot_formula, q_vector=q)
list_of_ranges = config.get_list_of_ranges(cable.k_vector[-1] + 1)
if cable.eval_cable_for_large_sigma(list_of_ranges, verbose=verbose, if cable.eval_cable_for_large_sigma(list_of_ranges, verbose=verbose,
print_results=print_results): print_results=print_results):
good_knots.append(cable) good_knots.append(cable.knot_description)
return good_knots return good_knots
# 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,
if limit is None: verbose=None, print_results=None):
limit = config.limit
if knot_formula is None: knot_formula, limit, verbose, print_results = \
knot_formula = config.knot_formula set_parameters(knot_formula, limit, verbose, print_results)
print_results = config.print_results
verbose = config.verbose
k_vector_size = extract_max(knot_formula) + 1 k_vector_size = extract_max(knot_formula) + 1
combinations = it.combinations_with_replacement(range(1, limit + 1), combinations = it.combinations_with_replacement(range(1, limit + 1),
@ -210,11 +163,57 @@ def search_for_null_signature_value(knot_formula=None, limit=None):
str(all_comb) + "\n") str(all_comb) + "\n")
f_results.write(line) f_results.write(line)
# searching for signature > 5 + #(v_i != 0) over given knot schema
def search_for_large_signature_value(knot_formula=None, limit=None,
verbose=None, print_results=None):
knot_formula, limit, verbose, print_results = \
set_parameters(knot_formula, limit, verbose, print_results)
k_vector_size = extract_max(knot_formula) + 1
combinations = it.combinations(range(1, limit + 1), k_vector_size)
P = Primes()
good_knots = []
# iterate over q-vector
for c in combinations:
q = [P.unrank(i + config.start_shift) for i in c]
if config.only_slice_candidates:
if not (q[3] > 4 * q[2] and
q[2] > 4 * q[1] and
q[1] > 4 * q[0]):
if verbose:
print("Ratio-condition does not hold")
continue
cable = TorusCable(knot_formula=knot_formula, q_vector=q)
list_of_ranges = config.get_list_of_ranges(cable.q_vector[-1])
if cable.eval_cable_for_large_signature(list_of_ranges, verbose=verbose,
print_results=print_results):
good_knots.append(cable.knot_description)
return good_knots
def get_shifted_combination(combination):
# for now applicable only for schama
# "[[k[0], k[1], k[2]], [k[3], k[4]],
# [-k[0], -k[3], -k[4]], [-k[1], -k[2]]]"
# shift the combination so that the knot can be a candidate for slice
combination = [combination[0], 4 * combination[0] + combination[1],
4 * (4 * combination[0] + combination[1]) + combination[2],
4 * combination[0] + combination[3],
4 * (4 * combination[0] + combination[3]) + combination[4]]
return combination
def extract_max(string): def extract_max(string):
numbers = re.findall('\d+', string) numbers = re.findall('\d+', string)
numbers = map(int, numbers) numbers = map(int, numbers)
return max(numbers) return max(numbers)
def is_trivial_combination(knot_sum): def is_trivial_combination(knot_sum):
# for now is applicable only for schema that are sums of 4 cables # for now is applicable only for schema that are sums of 4 cables
if len(knot_sum) == 4: if len(knot_sum) == 4: