functions searching for large signature values
This commit is contained in:
Maria Marchwicka
parent
2477c79297
commit
90322e0d19
@ -1,37 +1,6 @@
|
||||
#!/usr/bin/python
|
||||
|
||||
# TBD: read about Factory Method, variable in docstring, sage documentation
|
||||
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)
|
||||
|
||||
|
||||
"""
|
||||
This script calculates signature functions for knots (cable sums).
|
||||
@ -45,9 +14,9 @@ corresponds to a cable knot, e.g. a list
|
||||
T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7).
|
||||
|
||||
To calculate the number of characters for which signature function vanish use
|
||||
the function eval_cable_for_thetas as shown below.
|
||||
the function eval_cable_for_null_signature as shown below.
|
||||
|
||||
sage: eval_cable_for_thetas([[1, 3], [2], [-1, -2], [-3]])
|
||||
sage: eval_cable_for_null_signature([[1, 3], [2], [-1, -2], [-3]])
|
||||
|
||||
T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)
|
||||
Zero cases: 1
|
||||
@ -57,12 +26,24 @@ Zero theta combinations:
|
||||
|
||||
sage:
|
||||
|
||||
|
||||
The numbers given to the function eval_cable_for_thetas are k-values for each
|
||||
The numbers given to the function eval_cable_for_null_signature are k-values for each
|
||||
component/cable in a direct sum.
|
||||
|
||||
To calculate signature function for a knot and a theta value, use function
|
||||
get_function_of_theta_for_sum (see help/docstring for details).
|
||||
|
||||
About notation:
|
||||
Cables that we work with follow a schema:
|
||||
T(2, q_0; 2, q_1; 2, q_3) # -T(2, q_1; 2, q_3) #
|
||||
# T(2, q_2; 2, q_3) # -T(2, q_0; 2, q_2; 2, q_3)
|
||||
In knot_formula each k[i] is related with some q_i value, where
|
||||
q_i = 2*k[i] + 1.
|
||||
So we can work in the following steps:
|
||||
1) choose a schema/formula by changing the value of knot_formula
|
||||
2) set each q_i all or choose range in which q_i should varry
|
||||
3) choose vector v / theata vector.
|
||||
|
||||
|
||||
"""
|
||||
|
||||
import os
|
||||
@ -72,6 +53,7 @@ import collections
|
||||
import inspect
|
||||
import itertools as it
|
||||
import pandas as pd
|
||||
import numpy as np
|
||||
import re
|
||||
|
||||
|
||||
@ -81,30 +63,23 @@ class MySettings(object):
|
||||
|
||||
# is the ratio restriction for values in k_vector taken into account
|
||||
# False flag is usefull to make quick script tests
|
||||
self.only_slice_candidates = True
|
||||
# self.only_slice_candidates = False
|
||||
# self.only_slice_candidates = True
|
||||
self.only_slice_candidates = False
|
||||
|
||||
# knot_sum_formula is a schema for knots which signature function
|
||||
# knot_formula is a schema for knots which signature function
|
||||
# will be calculated
|
||||
self.knot_sum_formula = "[[k[0], k[1], k[2]], [k[3], k[4]], \
|
||||
[-k[0], -k[3], -k[4]], [-k[1], -k[2]]]"
|
||||
"""
|
||||
About notation:
|
||||
Cables that we work with follow a schema:
|
||||
T(2, q_0; 2, q_1; 2, q_2) # T(2, q_1; 2, q_2) #
|
||||
# -T(2, q_3; 2, q_2) # -T(2, q_0; 2, q_3; 2, q_2)
|
||||
In knot_sum_formula each k[i] is related with some q_i value, where
|
||||
q_i = 2*k[i] + 1.
|
||||
So we can work in the following steps:
|
||||
1) choose a schema/formula by changing the value of knot_sum_formula
|
||||
2) set each q_i all or choose range in which q_i should varry
|
||||
3) choose vector v / theata vector.
|
||||
self.knot_formula = "[[k[0], k[1], k[3]], [-k[1], -k[3]], \
|
||||
[k[2], k[3]], [-k[0], -k[2], -k[3]]]"
|
||||
|
||||
"""
|
||||
# self.knot_sum_formula = "[[k[0], k[1], k[2]], [k[3]],\
|
||||
|
||||
# self.knot_formula = "[[k[0], k[1], k[2]], [k[3], k[4]], \
|
||||
# [-k[0], -k[3], -k[4]], [-k[1], -k[2]]]"
|
||||
# self.knot_formula = "[[k[0], k[1], k[2]], [k[3]],\
|
||||
# [-k[0], -k[1], -k[3]], [-k[2]]]"
|
||||
self.default_limit = 3
|
||||
|
||||
self.verbose = True
|
||||
|
||||
|
||||
class SignatureFunction(object):
|
||||
"""
|
||||
@ -190,72 +165,200 @@ class SignatureFunction(object):
|
||||
def __str__(self):
|
||||
return ''.join([str(jump_arg) + ": " + str(jump) + "\n"
|
||||
for jump_arg, jump in sorted(self.data.items())])
|
||||
def value(self, arg):
|
||||
# Compute the value of the signature function at the point arg.
|
||||
# This requires summing all signature jumps that occur before arg.
|
||||
assert 0 <= arg and arg < 1
|
||||
val = 0
|
||||
for jump_arg, jump in self.data.items():
|
||||
if jump_arg < arg:
|
||||
val += 2 * jump
|
||||
elif jump_arg == arg:
|
||||
val += jump
|
||||
return val
|
||||
|
||||
|
||||
def main(arg):
|
||||
"""
|
||||
This function is run if the script was called from the terminal.
|
||||
It calls another function, perform_calculations,
|
||||
to calculate signature functions for a schema
|
||||
of a cable sum defined in the class MySettings.
|
||||
Optionaly a parameter (a limit for k_0 value) can be given.
|
||||
Thought to be run for time consuming calculations.
|
||||
"""
|
||||
try:
|
||||
new_limit = int(arg[1])
|
||||
except:
|
||||
new_limit = None
|
||||
perform_calculations(limit=new_limit)
|
||||
search_for_large_signature_value(limit=new_limit)
|
||||
# search_for_null_signature_value(limit=new_limit)
|
||||
|
||||
|
||||
def perform_calculations(knot_sum_formula=None, limit=None):
|
||||
"""
|
||||
This function calculates signature functions for knots constracted
|
||||
accordinga a schema for a cable sum. The schema is given as an argument
|
||||
or defined in the class MySettings.
|
||||
Results of calculations will be writen to a file and the stdout.
|
||||
limit is the upper bound for the first value in k_vector,
|
||||
i.e k[0] value in a cable sum, where q_0 = 2 * k[0] + 1.
|
||||
|
||||
(the number of knots that will be constracted depends on limit value).
|
||||
For each knot/cable sum the function eval_cable_for_thetas is called.
|
||||
eval_cable_for_thetas calculetes the number of all possible thetas
|
||||
(characters) and the number of combinations for which signature function
|
||||
equeles zero. In case the first number is larger than squere of the second,
|
||||
eval_cable_for_thetas returns None (i.e. the knot can not be slice).
|
||||
Data for knots that are candidates for slice knots are saved to a file.
|
||||
"""
|
||||
|
||||
settings = MySettings()
|
||||
def search_for_large_signature_value(knot_formula=None,
|
||||
limit=None, verbose=None):
|
||||
if limit is None:
|
||||
limit = settings.default_limit
|
||||
if knot_sum_formula is None:
|
||||
knot_sum_formula = settings.knot_sum_formula
|
||||
limit = config.default_limit
|
||||
if knot_formula is None:
|
||||
knot_formula = config.knot_formula
|
||||
if verbose is None:
|
||||
vebose = config.verbose
|
||||
|
||||
k_vector_size = extract_max(knot_sum_formula) + 1
|
||||
combinations = it.combinations_with_replacement(range(1, limit + 1),
|
||||
k_vector_size)
|
||||
|
||||
with open(settings.f_results, 'w') as f_results:
|
||||
k_vector_size = extract_max(knot_formula) + 1
|
||||
limit = max(limit, k_vector_size)
|
||||
print limit
|
||||
combinations = it.combinations(range(1, limit + 1), k_vector_size)
|
||||
for k in combinations:
|
||||
print
|
||||
print k
|
||||
# P = Primes()
|
||||
# sage: P.unrank(0)
|
||||
# 2
|
||||
# sage: P.unrank(5)
|
||||
# 13
|
||||
# sage: P.unrank(42)
|
||||
return None
|
||||
with open(config.f_results, 'w') as f_results:
|
||||
for k in combinations:
|
||||
# print
|
||||
# print k
|
||||
if verbose:
|
||||
print
|
||||
print k
|
||||
# TBD: maybe the following condition or the function
|
||||
# get_shifted_combination should be redefined to a dynamic version
|
||||
if settings.only_slice_candidates and k_vector_size == 5:
|
||||
if config.only_slice_candidates and k_vector_size == 5:
|
||||
k = get_shifted_combination(k)
|
||||
# print k
|
||||
knot_sum = eval(knot_sum_formula)
|
||||
knot_sum = eval(knot_formula)
|
||||
print "knot_sum"
|
||||
print knot_sum
|
||||
|
||||
if is_trivial_combination(knot_sum):
|
||||
if verbose:
|
||||
print "\nTrivial combination" + str(knot_sum)
|
||||
continue
|
||||
result = eval_cable_for_thetas(knot_sum, print_results=False)
|
||||
result = eval_cable_for_large_signature(knot_sum,
|
||||
print_results=False)
|
||||
if result is not None:
|
||||
knot_description, null_comb, all_comb = result
|
||||
line = (str(k) + ", " + str(null_comb) + ", " +
|
||||
str(all_comb) + "\n")
|
||||
f_results.write(line)
|
||||
|
||||
def eval_cable_for_large_signature(knot_sum, print_results=True, verbose=None):
|
||||
if verbose is None:
|
||||
verbose = config.verbose
|
||||
if verbose:
|
||||
print "\neval_cable_for_large_signature"
|
||||
if len(knot_sum) != 4:
|
||||
print "Wrong number of cable direct summands!"
|
||||
return None
|
||||
q = 2 * abs(knot_sum[-1][-1]) + 1
|
||||
print "q is " + str(q)
|
||||
|
||||
f = get_function_of_theta_for_sum(*knot_sum, verbose=False)
|
||||
knot_description = get_knot_descrption(*knot_sum)
|
||||
all_combinations = get_number_of_combinations(*knot_sum)
|
||||
|
||||
null_combinations = 0
|
||||
zero_theta_combinations = []
|
||||
|
||||
ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum]
|
||||
if verbose:
|
||||
print "eval_cable_for_large_signature - knot_description: "
|
||||
print knot_description
|
||||
for v_theta in it.product(*ranges_list):
|
||||
y = f(*v_theta).value(1/2)
|
||||
print y
|
||||
if abs(y) > 5 + np.count_nonzero(v_theta):
|
||||
print "hura hura"
|
||||
# == 0:
|
||||
# zero_theta_combinations.append(v_theta)
|
||||
# m = len([theta for theta in v_theta if theta != 0])
|
||||
# null_combinations += 2^m
|
||||
# else:
|
||||
# assert sum(v_theta) != 0
|
||||
|
||||
if print_results:
|
||||
print
|
||||
print knot_description
|
||||
print "Zero cases: " + str(null_combinations)
|
||||
print "All cases: " + str(all_combinations)
|
||||
if zero_theta_combinations:
|
||||
print "Zero theta combinations: "
|
||||
for el in zero_theta_combinations:
|
||||
print el
|
||||
if null_combinations^2 >= all_combinations:
|
||||
return knot_description, null_combinations, all_combinations
|
||||
return None
|
||||
|
||||
|
||||
def search_for_null_signature_value(knot_formula=None, limit=None):
|
||||
if limit is None:
|
||||
limit = config.default_limit
|
||||
if knot_formula is None:
|
||||
knot_formula = config.knot_formula
|
||||
|
||||
k_vector_size = extract_max(knot_formula) + 1
|
||||
combinations = it.combinations_with_replacement(range(1, limit + 1),
|
||||
k_vector_size)
|
||||
|
||||
with open(config.f_results, 'w') as f_results:
|
||||
for k in combinations:
|
||||
# print
|
||||
# print k
|
||||
# TBD: maybe the following condition or the function
|
||||
# get_shifted_combination should be redefined to a dynamic version
|
||||
if confi.only_slice_candidates and k_vector_size == 5:
|
||||
k = get_shifted_combination(k)
|
||||
# print k
|
||||
knot_sum = eval(knot_formula)
|
||||
|
||||
if is_trivial_combination(knot_sum):
|
||||
continue
|
||||
result = search_for_thetas(knot_sum, print_results=False)
|
||||
if result is not None:
|
||||
knot_description, null_comb, all_comb = result
|
||||
line = (str(k) + ", " + str(null_comb) + ", " +
|
||||
str(all_comb) + "\n")
|
||||
f_results.write(line)
|
||||
|
||||
def search_for_thetas(knot_sum, print_results=False, verbose=None):
|
||||
if verbose is None:
|
||||
vebose = confi.verbose
|
||||
if verbose:
|
||||
print "search_for_thetas"
|
||||
f = get_function_of_theta_for_sum(*knot_sum)
|
||||
knot_description = get_knot_descrption(*knot_sum)
|
||||
all_combinations = get_number_of_combinations(*knot_sum)
|
||||
|
||||
large_value_combinations = 0
|
||||
good_theta_list = []
|
||||
|
||||
ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum]
|
||||
if verbose:
|
||||
print "knot_description:"
|
||||
print knot_description
|
||||
for v_theta in it.product(*ranges_list):
|
||||
if (v_theta[0]^2 - v_theta[1]^2 + v_theta[2]^2 - v_theta[3]^2) % q:
|
||||
print "ojojoj"
|
||||
print (v_theta[0]^2 - v_theta[1]^2 + v_theta[2]^2 - v_theta[3]^2) % q
|
||||
continue
|
||||
print f(*v_theta).value(-1)
|
||||
if f(*v_theta).value(-1):
|
||||
print "Hip hip hura"
|
||||
good_theta_list.append(v_theta)
|
||||
m = len([theta for theta in v_theta if theta != 0])
|
||||
large_value_combinations += 2^m
|
||||
else:
|
||||
print "smuteczek"
|
||||
# else:
|
||||
# assert sum(v_theta) != 0
|
||||
|
||||
if print_results:
|
||||
print
|
||||
print knot_description
|
||||
print "Zero cases: " + str(null_combinations)
|
||||
print "All cases: " + str(all_combinations)
|
||||
if zero_theta_combinations:
|
||||
print "Zero theta combinations: "
|
||||
for el in zero_theta_combinations:
|
||||
print el
|
||||
if null_combinations^2 >= all_combinations:
|
||||
return knot_description, null_combinations, all_combinations
|
||||
return None
|
||||
|
||||
|
||||
def is_trivial_combination(knot_sum):
|
||||
# for now is applicable only for schema that are sums of 4 cables
|
||||
@ -279,19 +382,6 @@ def get_shifted_combination(combination):
|
||||
|
||||
|
||||
def get_blanchfield_for_pattern(k_n, theta):
|
||||
"""
|
||||
Arguments:
|
||||
k_n: a number s.t. q_n = 2 * k_n + 1, where
|
||||
T(2, q_n) is a pattern knot for a single cable from a cable sum
|
||||
theta: twist/character for the cable (value form v vector)
|
||||
Return:
|
||||
SignatureFunction created for twisted signature function
|
||||
for a given cable and theta/character
|
||||
Based on:
|
||||
Proposition 9.8. in Twisted Blanchfield Pairing
|
||||
(https://arxiv.org/pdf/1809.08791.pdf)
|
||||
"""
|
||||
|
||||
if theta == 0:
|
||||
a = get_untwisted_signature_function(k_n)
|
||||
return a.square_root() + a.minus_square_root()
|
||||
@ -322,28 +412,7 @@ def get_blanchfield_for_pattern(k_n, theta):
|
||||
|
||||
|
||||
def get_cable_signature_as_theta_function(*arg):
|
||||
"""
|
||||
Argument:
|
||||
n integers that encode a single cable, i.e.
|
||||
values of q_i for T(2,q_0; 2,q_1; ... 2, q_n)
|
||||
Return:
|
||||
a function that returns SignatureFunction for this single cable
|
||||
and a theta given as an argument
|
||||
"""
|
||||
|
||||
def get_signture_function(theta):
|
||||
"""
|
||||
This function returns SignatureFunction for previously defined single
|
||||
cable T_(2, q) and a theta given as an argument.
|
||||
The cable was defined by calling function
|
||||
get_cable_signature_as_theta_function(*arg)
|
||||
with the cable description as an argument.
|
||||
It is an implementaion of the formula:
|
||||
Bl_theta(K'_(2, d)) =
|
||||
Bl_theta(T_2, d) + Bl(K')(ksi_l^(-theta) * t)
|
||||
+ Bl(K')(ksi_l^theta * t)
|
||||
"""
|
||||
|
||||
# TBD: another formula (for t^2) description
|
||||
|
||||
k_n = abs(arg[-1])
|
||||
@ -368,6 +437,7 @@ def get_cable_signature_as_theta_function(*arg):
|
||||
c = c.double_cover()
|
||||
cable_signature += b + c
|
||||
return cable_signature
|
||||
get_signture_function.__doc__ = get_signture_function_docsting
|
||||
return get_signture_function
|
||||
|
||||
|
||||
@ -381,7 +451,232 @@ def get_untwisted_signature_function(j):
|
||||
return SignatureFunction(w)
|
||||
|
||||
|
||||
def get_function_of_theta_for_sum(*arg):
|
||||
def get_function_of_theta_for_sum(*arg, **key_args):
|
||||
if 'verbose' in key_args:
|
||||
verbose_default = key_args['verbose']
|
||||
else:
|
||||
verbose_default = confi.verbose
|
||||
def signature_function_for_sum(*thetas, **kwargs):
|
||||
verbose = verbose_default
|
||||
if 'verbose' in kwargs:
|
||||
verbose = kwargs['verbose']
|
||||
la = len(arg)
|
||||
lt = len(thetas)
|
||||
|
||||
# call with no arguments
|
||||
if lt == 0:
|
||||
return signature_function_for_sum(*(la * [0]))
|
||||
|
||||
if lt != la:
|
||||
msg = "This function takes exactly " + str(la) + \
|
||||
" arguments or no argument at all (" + str(lt) + " given)."
|
||||
raise TypeError(msg)
|
||||
|
||||
sf = SignatureFunction([(0, 0)])
|
||||
|
||||
# for each cable in cable sum apply theta
|
||||
for i, knot in enumerate(arg):
|
||||
try:
|
||||
sf += (get_cable_signature_as_theta_function(*knot))(thetas[i])
|
||||
# in case wrong theata value was given
|
||||
except ValueError as e:
|
||||
print "ValueError: " + str(e.args[0]) +\
|
||||
" Please change " + str(i + 1) + ". parameter."
|
||||
return None
|
||||
if verbose:
|
||||
print
|
||||
print str(thetas)
|
||||
print sf
|
||||
return sf
|
||||
signature_function_for_sum.__doc__ = signature_function_for_sum_docstring
|
||||
return signature_function_for_sum
|
||||
|
||||
|
||||
def eval_cable_for_null_signature(knot_sum, print_results=True, verbose=None):
|
||||
# search for zero combinations
|
||||
if verbose is None:
|
||||
vebose = confi.verbose
|
||||
f = get_function_of_theta_for_sum(*knot_sum, verbose=False)
|
||||
knot_description = get_knot_descrption(*knot_sum)
|
||||
all_combinations = get_number_of_combinations(*knot_sum)
|
||||
|
||||
null_combinations = 0
|
||||
zero_theta_combinations = []
|
||||
|
||||
ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum]
|
||||
if verbose:
|
||||
print
|
||||
print knot_description
|
||||
for v_theta in it.product(*ranges_list):
|
||||
if f(*v_theta, verbose=False).sum_of_absolute_values() == 0:
|
||||
zero_theta_combinations.append(v_theta)
|
||||
m = len([theta for theta in v_theta if theta != 0])
|
||||
null_combinations += 2^m
|
||||
# else:
|
||||
# assert sum(v_theta) != 0
|
||||
|
||||
if print_results:
|
||||
print
|
||||
print knot_description
|
||||
print "Zero cases: " + str(null_combinations)
|
||||
print "All cases: " + str(all_combinations)
|
||||
if zero_theta_combinations:
|
||||
print "Zero theta combinations: "
|
||||
for el in zero_theta_combinations:
|
||||
print el
|
||||
if null_combinations^2 >= all_combinations:
|
||||
return knot_description, null_combinations, all_combinations
|
||||
return None
|
||||
|
||||
|
||||
def check_squares(a, k):
|
||||
print
|
||||
p = 2 * k + 1
|
||||
k_0 = (p^2 - 1)/2
|
||||
knot_sum = [[a, k], [k_0], [-a, -k_0], [-k]]
|
||||
print get_knot_descrption(*knot_sum)
|
||||
if a * 4 >= p or is_trivial_combination(knot_sum):
|
||||
if a * 4 >= p:
|
||||
print str(knot_sum)
|
||||
print "a * 4 >= p"
|
||||
else:
|
||||
print "Trivial " + str(knot_sum)
|
||||
return None
|
||||
eval_cable_for_null_signature(knot_sum)
|
||||
|
||||
|
||||
def get_number_of_combinations(*arg):
|
||||
number_of_combinations = 1
|
||||
for knot in arg:
|
||||
number_of_combinations *= (2 * abs(knot[-1]) + 1)
|
||||
return number_of_combinations
|
||||
|
||||
|
||||
def extract_max(string):
|
||||
numbers = re.findall('\d+', string)
|
||||
numbers = map(int, numbers)
|
||||
return max(numbers)
|
||||
|
||||
|
||||
def mod_one(n):
|
||||
return n - floor(n)
|
||||
|
||||
|
||||
def get_knot_descrption(*arg):
|
||||
description = ""
|
||||
for knot in arg:
|
||||
if knot[0] < 0:
|
||||
description += "-"
|
||||
description += "T("
|
||||
for k in knot:
|
||||
description += "2, " + str(2 * abs(k) + 1) + "; "
|
||||
description = description[:-2] + ") # "
|
||||
return description[:-3]
|
||||
|
||||
get_blanchfield_for_pattern.__doc__ = \
|
||||
"""
|
||||
Arguments:
|
||||
k_n: a number s.t. q_n = 2 * k_n + 1, where
|
||||
T(2, q_n) is a pattern knot for a single cable from a cable sum
|
||||
theta: twist/character for the cable (value form v vector)
|
||||
Return:
|
||||
SignatureFunction created for twisted signature function
|
||||
for a given cable and theta/character
|
||||
Based on:
|
||||
Proposition 9.8. in Twisted Blanchfield Pairing
|
||||
(https://arxiv.org/pdf/1809.08791.pdf)
|
||||
"""
|
||||
|
||||
get_number_of_combinations.__doc__ = \
|
||||
"""
|
||||
Arguments:
|
||||
arbitrary number of lists of numbers, each list encodes a single cable
|
||||
Return:
|
||||
number of possible theta values combinations that could be applied
|
||||
for a given cable sum,
|
||||
i.e. the product of q_j for j = {1,.. n},
|
||||
where n is a number of direct components in the cable sum,
|
||||
and q_j is the last q parameter for the component (a single cable)
|
||||
"""
|
||||
|
||||
get_knot_descrption.__doc__ = \
|
||||
"""
|
||||
Arguments:
|
||||
arbitrary number of lists of numbers, each list encodes a single cable.
|
||||
Examples:
|
||||
sage: get_knot_descrption([1, 3], [2], [-1, -2], [-3])
|
||||
'T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)'
|
||||
"""
|
||||
|
||||
mod_one.__doc__ = \
|
||||
"""
|
||||
Argument:
|
||||
a number
|
||||
Return:
|
||||
the fractional part of the argument
|
||||
Examples:
|
||||
sage: mod_one(9 + 3/4)
|
||||
3/4
|
||||
sage: mod_one(-9 + 3/4)
|
||||
3/4
|
||||
sage: mod_one(-3/4)
|
||||
1/4
|
||||
"""
|
||||
|
||||
search_for_null_signature_value.__doc__ = \
|
||||
"""
|
||||
This function calculates signature functions for knots constracted
|
||||
accordinga a schema for a cable sum. The schema is given as an argument
|
||||
or defined in the class MySettings.
|
||||
Results of calculations will be writen to a file and the stdout.
|
||||
limit is the upper bound for the first value in k_vector,
|
||||
i.e k[0] value in a cable sum, where q_0 = 2 * k[0] + 1.
|
||||
|
||||
(the number of knots that will be constracted depends on limit value).
|
||||
For each knot/cable sum the function eval_cable_for_null_signature is called.
|
||||
eval_cable_for_null_signature calculetes the number of all possible thetas
|
||||
(characters) and the number of combinations for which signature function
|
||||
equeles zero. In case the first number is larger than squere of the second,
|
||||
eval_cable_for_null_signature returns None (i.e. the knot can not be slice).
|
||||
Data for knots that are candidates for slice knots are saved to a file.
|
||||
"""
|
||||
|
||||
extract_max.__doc__ = \
|
||||
"""
|
||||
Return:
|
||||
maximum of absolute values of numbers from given string
|
||||
Examples:
|
||||
sage: extract_max("([1, 3], [2], [-1, -2], [-10])")
|
||||
10
|
||||
sage: extract_max("3, 55, ewewe, -42, 3300, 50")
|
||||
3300
|
||||
"""
|
||||
|
||||
eval_cable_for_null_signature.__doc__ = \
|
||||
"""
|
||||
This function calculates all possible twisted signature functions for
|
||||
a knot that is given as an argument. The knot should be encoded as a list
|
||||
of its direct component. Each component schould be presented as a list
|
||||
of integers. This integers correspond to the k - values in each component/
|
||||
cable. If a component is a mirror image of a cable the minus sign should
|
||||
be written before each number for this component. For example:
|
||||
eval_cable_for_null_signature([[1, 8], [2], [-2, -8], [-2]])
|
||||
eval_cable_for_null_signature([[1, 2], [-1, -2]])
|
||||
|
||||
sage: eval_cable_for_null_signature([[1, 3], [2], [-1, -2], [-3]])
|
||||
|
||||
T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)
|
||||
Zero cases: 1
|
||||
All cases: 1225
|
||||
Zero theta combinations:
|
||||
(0, 0, 0, 0)
|
||||
|
||||
sage:
|
||||
The numbers given to the function eval_cable_for_null_signature are k-values for each
|
||||
component/cable in a direct sum.
|
||||
"""
|
||||
|
||||
get_function_of_theta_for_sum.__doc__ = \
|
||||
"""
|
||||
Function intended to construct signature function for a connected
|
||||
sum of multiple cables with varying theta parameter values.
|
||||
@ -435,193 +730,84 @@ def get_function_of_theta_for_sum(*arg):
|
||||
6/7: 0
|
||||
"""
|
||||
|
||||
def signature_function_for_sum(*thetas, **kwargs):
|
||||
"""
|
||||
Arguments:
|
||||
|
||||
Returns object of SignatureFunction class for a previously defined
|
||||
connected sum of len(arg) cables.
|
||||
Acept len(arg) arguments: for each cable one theta parameter.
|
||||
If call with no arguments, all theta parameters are set to be 0.
|
||||
"""
|
||||
if 'verbose' in kwargs:
|
||||
verbose = kwargs['verbose']
|
||||
else:
|
||||
verbose = False
|
||||
|
||||
la = len(arg)
|
||||
lt = len(thetas)
|
||||
|
||||
# call with no arguments
|
||||
if lt == 0:
|
||||
return signature_function_for_sum(*(la * [0]))
|
||||
|
||||
if lt != la:
|
||||
msg = "This function takes exactly " + str(la) + \
|
||||
" arguments or no argument at all (" + str(lt) + " given)."
|
||||
raise TypeError(msg)
|
||||
|
||||
sf = SignatureFunction([(0, 0)])
|
||||
|
||||
# for each cable in cable sum apply theta
|
||||
for i, knot in enumerate(arg):
|
||||
try:
|
||||
sf += (get_cable_signature_as_theta_function(*knot))(thetas[i])
|
||||
# in case wrong theata value was given
|
||||
except ValueError as e:
|
||||
print "ValueError: " + str(e.args[0]) +\
|
||||
" Please change " + str(i + 1) + ". parameter."
|
||||
return None
|
||||
if verbose:
|
||||
print
|
||||
print str(*thetas)
|
||||
print sf
|
||||
return sf
|
||||
return signature_function_for_sum
|
||||
|
||||
|
||||
def eval_cable_for_thetas(knot_sum, print_results=True, verbose=False):
|
||||
"""
|
||||
This function calculates all possible twisted signature functions for
|
||||
a knot that is given as an argument. The knot should be encoded as a list
|
||||
of its direct component. Each component schould be presented as a list
|
||||
of integers. This integers correspond to the k - values in each component/
|
||||
cable. If a component is a mirror image of a cable the minus sign should
|
||||
be written before each number for this component. For example:
|
||||
eval_cable_for_thetas([[1, 8], [2], [-2, -8], [-2]])
|
||||
eval_cable_for_thetas([[1, 2], [-1, -2]])
|
||||
|
||||
sage: eval_cable_for_thetas([[1, 3], [2], [-1, -2], [-3]])
|
||||
|
||||
T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)
|
||||
Zero cases: 1
|
||||
All cases: 1225
|
||||
Zero theta combinations:
|
||||
(0, 0, 0, 0)
|
||||
|
||||
sage:
|
||||
The numbers given to the function eval_cable_for_thetas are k-values for each
|
||||
component/cable in a direct sum.
|
||||
|
||||
|
||||
"""
|
||||
f = get_function_of_theta_for_sum(*knot_sum)
|
||||
knot_description = get_knot_descrption(*knot_sum)
|
||||
all_combinations = get_number_of_combinations(*knot_sum)
|
||||
|
||||
null_combinations = 0
|
||||
zero_theta_combinations = []
|
||||
|
||||
ranges_list = [range(abs(knot[-1]) + 1) for knot in knot_sum]
|
||||
if verbose:
|
||||
print
|
||||
print knot_description
|
||||
for v_theta in it.product(*ranges_list):
|
||||
if f(*v_theta, verbose=verbose).sum_of_absolute_values() == 0:
|
||||
zero_theta_combinations.append(v_theta)
|
||||
m = len([theta for theta in v_theta if theta != 0])
|
||||
null_combinations += 2^m
|
||||
# else:
|
||||
# assert sum(v_theta) != 0
|
||||
|
||||
if print_results:
|
||||
print
|
||||
print knot_description
|
||||
print "Zero cases: " + str(null_combinations)
|
||||
print "All cases: " + str(all_combinations)
|
||||
if zero_theta_combinations:
|
||||
print "Zero theta combinations: "
|
||||
for el in zero_theta_combinations:
|
||||
print el
|
||||
if null_combinations^2 >= all_combinations:
|
||||
return knot_description, null_combinations, all_combinations
|
||||
return None
|
||||
|
||||
|
||||
def check_squares(a, k):
|
||||
print
|
||||
p = 2 * k + 1
|
||||
k_0 = (p^2 - 1)/2
|
||||
knot_sum = [[a, k], [k_0], [-a, -k_0], [-k]]
|
||||
print get_knot_descrption(*knot_sum)
|
||||
if a * 4 >= p or is_trivial_combination(knot_sum):
|
||||
if a * 4 >= p:
|
||||
print str(knot_sum)
|
||||
print "a * 4 >= p"
|
||||
else:
|
||||
print "Trivial " + str(knot_sum)
|
||||
return None
|
||||
eval_cable_for_thetas(knot_sum)
|
||||
|
||||
|
||||
def get_number_of_combinations(*arg):
|
||||
"""
|
||||
Arguments:
|
||||
arbitrary number of lists of numbers, each list encodes a single cable
|
||||
Return:
|
||||
number of possible theta values combinations that could be applied
|
||||
for a given cable sum,
|
||||
i.e. the product of q_j for j = {1,.. n},
|
||||
where n is a number of direct components in the cable sum,
|
||||
and q_j is the last q parameter for the component (a single cable)
|
||||
"""
|
||||
number_of_combinations = 1
|
||||
for knot in arg:
|
||||
number_of_combinations *= (2 * abs(knot[-1]) + 1)
|
||||
return number_of_combinations
|
||||
|
||||
|
||||
def extract_max(string):
|
||||
"""
|
||||
Return:
|
||||
maximum of absolute values of numbers from given string
|
||||
Examples:
|
||||
sage: extract_max("([1, 3], [2], [-1, -2], [-10])")
|
||||
10
|
||||
sage: extract_max("3, 55, ewewe, -42, 3300, 50")
|
||||
3300
|
||||
"""
|
||||
numbers = re.findall('\d+', string)
|
||||
numbers = map(int, numbers)
|
||||
return max(numbers)
|
||||
|
||||
|
||||
def mod_one(n):
|
||||
get_cable_signature_as_theta_function.__doc__ = \
|
||||
"""
|
||||
Argument:
|
||||
a number
|
||||
n integers that encode a single cable, i.e.
|
||||
values of q_i for T(2,q_0; 2,q_1; ... 2, q_n)
|
||||
Return:
|
||||
the fractional part of the argument
|
||||
Examples:
|
||||
sage: mod_one(9 + 3/4)
|
||||
3/4
|
||||
sage: mod_one(-9 + 3/4)
|
||||
3/4
|
||||
sage: mod_one(-3/4)
|
||||
1/4
|
||||
a function that returns SignatureFunction for this single cable
|
||||
and a theta given as an argument
|
||||
"""
|
||||
return n - floor(n)
|
||||
|
||||
get_signture_function_docsting = \
|
||||
"""
|
||||
This function returns SignatureFunction for previously defined single
|
||||
cable T_(2, q) and a theta given as an argument.
|
||||
The cable was defined by calling function
|
||||
get_cable_signature_as_theta_function(*arg)
|
||||
with the cable description as an argument.
|
||||
It is an implementaion of the formula:
|
||||
Bl_theta(K'_(2, d)) =
|
||||
Bl_theta(T_2, d) + Bl(K')(ksi_l^(-theta) * t)
|
||||
+ Bl(K')(ksi_l^theta * t)
|
||||
"""
|
||||
|
||||
def get_knot_descrption(*arg):
|
||||
signature_function_for_sum_docstring = \
|
||||
"""
|
||||
Arguments:
|
||||
arbitrary number of lists of numbers, each list encodes a single cable.
|
||||
Examples:
|
||||
sage: get_knot_descrption([1, 3], [2], [-1, -2], [-3])
|
||||
'T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)'
|
||||
|
||||
Returns object of SignatureFunction class for a previously defined
|
||||
connected sum of len(arg) cables.
|
||||
Acept len(arg) arguments: for each cable one theta parameter.
|
||||
If call with no arguments, all theta parameters are set to be 0.
|
||||
"""
|
||||
description = ""
|
||||
for knot in arg:
|
||||
if knot[0] < 0:
|
||||
description += "-"
|
||||
description += "T("
|
||||
for k in knot:
|
||||
description += "2, " + str(2 * abs(k) + 1) + "; "
|
||||
description = description[:-2] + ") # "
|
||||
return description[:-3]
|
||||
|
||||
main.__doc__ = \
|
||||
"""
|
||||
This function is run if the script was called from the terminal.
|
||||
It calls another function, search_for_null_signature_value,
|
||||
to calculate signature functions for a schema
|
||||
of a cable sum defined in the class MySettings.
|
||||
Optionaly a parameter (a limit for k_0 value) can be given.
|
||||
Thought to be run for time consuming calculations.
|
||||
"""
|
||||
|
||||
config = MySettings()
|
||||
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)
|
||||
#
|
||||
#
|
||||
|
||||
Loading…
Reference in New Issue
Block a user