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classes as separate module - now working

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Maria Marchwicka 2020-08-31 18:12:47 +02:00
parent 0110e6161c
commit d9986b63c1
2 changed files with 312 additions and 270 deletions
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291
cable_signature.sage
View File

@ -1,6 +1,8 @@
#!/usr/bin/python
import collections
import numpy as np
import itertools as it
import re
class TorusCable(object):
def __init__(self, knot_formula, k_vector=None, q_vector=None):
@ -514,3 +516,290 @@ class SignatureFunction(object):
def mod_one(n):
return n - floor(n)
def get_untwisted_signature_function(j):
# return the signature function of the T_{2,2k+1} torus knot
k = abs(j)
w = ([((2 * a + 1)/(4 * k + 2), -1 * sgn(j)) for a in range(k)] +
[((2 * a + 1)/(4 * k + 2), 1 * sgn(j))
for a in range(k + 1, 2 * k + 1)])
return SignatureFunction(values=w)
def get_signature_summand_as_theta_function(*arg):
def get_signture_function(theta):
# TBD: another formula (for t^2) description
k_n = abs(arg[-1])
if theta > k_n:
msg = "k for the pattern in the cable is " + str(arg[-1]) + \
". Parameter theta should not be larger than abs(k)."
raise ValueError(msg)
# twisted part
cable_signature = get_blanchfield_for_pattern(arg[-1], theta)
# untwisted part
for i, k in enumerate(arg[:-1][::-1]):
ksi = 1/(2 * k_n + 1)
power = 2^i
a = get_untwisted_signature_function(k)
shift = theta * ksi * power
b = a >> shift
c = a << shift
for _ in range(i):
b = b.double_cover()
c = c.double_cover()
cable_signature += b + c
test = b - c
test2 = -c + b
assert test == test
return cable_signature
get_signture_function.__doc__ = get_signture_function_docsting
return get_signture_function
def get_blanchfield_for_pattern(k_n, theta):
if theta == 0:
a = get_untwisted_signature_function(k_n)
return a.square_root() + a.minus_square_root()
results = []
k = abs(k_n)
ksi = 1/(2 * k + 1)
# lambda_odd, i.e. (theta + e) % 2 != 0
for e in range(1, k + 1):
if (theta + e) % 2 != 0:
results.append((e * ksi, 1 * sgn(k_n)))
results.append((1 - e * ksi, -1 * sgn(k_n)))
# lambda_even
# print("normal")
for e in range(1, theta):
if (theta + e) % 2 == 0:
results.append((e * ksi, 1 * sgn(k_n)))
results.append((1 - e * ksi, -1 * sgn(k_n)))
# print("reversed")
for e in range(theta + 1, k + 1):
if (theta + e) % 2 != 0:
continue
results.append((e * ksi, -1 * sgn(k_n)))
results.append((1 - e * ksi, 1 * sgn(k_n)))
return SignatureFunction(values=results)
def get_signature_summand_as_theta_function(*arg):
def get_signture_function(theta):
# TBD: another formula (for t^2) description
k_n = abs(arg[-1])
if theta > k_n:
msg = "k for the pattern in the cable is " + str(arg[-1]) + \
". Parameter theta should not be larger than abs(k)."
raise ValueError(msg)
# twisted part
cable_signature = get_blanchfield_for_pattern(arg[-1], theta)
# untwisted part
for i, k in enumerate(arg[:-1][::-1]):
ksi = 1/(2 * k_n + 1)
power = 2^i
a = get_untwisted_signature_function(k)
shift = theta * ksi * power
b = a >> shift
c = a << shift
for _ in range(i):
b = b.double_cover()
c = c.double_cover()
cable_signature += b + c
test = b - c
test2 = -c + b
assert test == test
return cable_signature
get_signture_function.__doc__ = get_signture_function_docsting
return get_signture_function
def get_untwisted_signature_function(j):
# return the signature function of the T_{2,2k+1} torus knot
k = abs(j)
w = ([((2 * a + 1)/(4 * k + 2), -1 * sgn(j)) for a in range(k)] +
[((2 * a + 1)/(4 * k + 2), 1 * sgn(j))
for a in range(k + 1, 2 * k + 1)])
return SignatureFunction(values=w)
TorusCable.get_number_of_combinations_of_theta.__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)
"""
TorusCable.get_knot_descrption.__doc__ = \
"""
Arguments:
arbitrary number of lists of numbers, each list encodes a single cable.
Examples:
sage: get_knot_descrption([1, 3], [2], [-1, -2], [-3])
'T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)'
"""
TorusCable.eval_cable_for_null_signature.__doc__ = \
"""
This function calculates all possible twisted signature functions for
a knot that is given as an argument. The knot should be encoded as a list
of its direct component. Each component schould be presented as a list
of integers. This integers correspond to the k - values in each component/
cable. If a component is a mirror image of a cable the minus sign should
be written before each number for this component. For example:
eval_cable_for_null_signature([[1, 8], [2], [-2, -8], [-2]])
eval_cable_for_null_signature([[1, 2], [-1, -2]])
sage: eval_cable_for_null_signature([[1, 3], [2], [-1, -2], [-3]])
T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)
Zero cases: 1
All cases: 1225
Zero theta combinations:
(0, 0, 0, 0)
sage:
The numbers given to the function eval_cable_for_null_signature
are k-values for each component/cable in a direct sum.
"""
TorusCable.get_signature_as_function_of_theta.__doc__ = \
"""
Function intended to construct signature function for a connected
sum of multiple cables with varying theta parameter values.
Accept arbitrary number of arguments (depending on number of cables in
connected sum).
Each argument should be given as list of integer representing
k - parameters for a cable: parameters k_i (i=1,.., n-1) for satelit knots
T(2, 2k_i + 1) and - the last one - k_n for a pattern knot T(2, 2k_n + 1).
Returns a function that will take theta vector as an argument and return
an object SignatureFunction.
To calculate signature function for a cable sum and a theta values vector,
use as below.
sage: signature_function_generator = get_signature_as_function_of_theta(
[1, 3], [2], [-1, -2], [-3])
sage: sf = signature_function_generator(2, 1, 2, 2)
sage: print(sf)
0: 0
5/42: 1
1/7: 0
1/5: -1
7/30: -1
2/5: 1
3/7: 0
13/30: -1
19/42: -1
23/42: 1
17/30: 1
4/7: 0
3/5: -1
23/30: 1
4/5: 1
6/7: 0
37/42: -1
Or like below.
sage: print(get_signature_as_function_of_theta([1, 3], [2], [-1, -2], [-3]
)(2, 1, 2, 2))
0: 0
1/7: 0
1/6: 0
1/5: -1
2/5: 1
3/7: 0
1/2: 0
4/7: 0
3/5: -1
4/5: 1
5/6: 0
6/7: 0
"""
SignatureFunction.__doc__ = \
"""
This simple class encodes twisted and untwisted signature functions
of knots. Since the signature function is entirely encoded by its signature
jump, the class stores only information about signature jumps
in a dictionary self.cnt_signature_jumps.
The dictionary stores data of the signature jump as a key/values pair,
where the key is the argument at which the functions jumps
and value encodes the value of the jump. Remember that we treat
signature functions as defined on the interval [0,1).
"""
get_signture_function_docsting = \
"""
This function returns SignatureFunction for previously defined single
cable T_(2, q) and a theta given as an argument.
The cable was defined by calling function
get_signature_summand_as_theta_function(*arg)
with the cable description as an argument.
It is an implementaion of the formula:
Bl_theta(K'_(2, d)) =
Bl_theta(T_2, d) + Bl(K')(ksi_l^(-theta) * t)
+ Bl(K')(ksi_l^theta * t)
"""
signature_as_function_of_theta_docstring = \
"""
Arguments:
Returns object of SignatureFunction class for a previously defined
connected sum of len(arg) cables.
Accept len(arg) arguments: for each cable one theta parameter.
If call with no arguments, all theta parameters are set to be 0.
"""
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
"""
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_signature_summand_as_theta_function.__doc__ = \
"""
Argument:
n integers that encode a single cable, i.e.
values of q_i for T(2,q_0; 2,q_1; ... 2, q_n)
Return:
a function that returns SignatureFunction for this single cable
and a theta given as an argument
"""

291
my_signature.sage
View File

@ -11,6 +11,11 @@ import collections
import itertools as it
import numpy as np
import re
# try:
# from cable_signature import SignatureFunction, TorusCable
# except ModuleNotFoundError:
os.system('sage --preparse cable_signature.sage')
os.system('mv cable_signature.sage.py cable_signature.py')
from cable_signature import SignatureFunction, TorusCable
class Config(object):
@ -189,7 +194,7 @@ def search_for_null_signature_value(knot_formula=None, limit=None):
for k in combinations:
if config.only_slice_candidates and k_vector_size == 5:
k = get_shifted_combination(k)
cable = TorusCable(knot_formula, k_vector=k)
cable = TorusCable(knot_formula, k_vector=k)
if is_trivial_combination(cable.knot_sum):
print(cable.knot_sum)
continue
@ -201,6 +206,11 @@ def search_for_null_signature_value(knot_formula=None, limit=None):
str(all_comb) + "\n")
f_results.write(line)
def extract_max(string):
numbers = re.findall('\d+', string)
numbers = map(int, numbers)
return max(numbers)
def is_trivial_combination(knot_sum):
# for now is applicable only for schema that are sums of 4 cables
if len(knot_sum) == 4:
@ -209,144 +219,6 @@ def is_trivial_combination(knot_sum):
return True
return False
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 get_blanchfield_for_pattern(k_n, theta):
if theta == 0:
a = get_untwisted_signature_function(k_n)
return a.square_root() + a.minus_square_root()
results = []
k = abs(k_n)
ksi = 1/(2 * k + 1)
# lambda_odd, i.e. (theta + e) % 2 != 0
for e in range(1, k + 1):
if (theta + e) % 2 != 0:
results.append((e * ksi, 1 * sgn(k_n)))
results.append((1 - e * ksi, -1 * sgn(k_n)))
# lambda_even
# print("normal")
for e in range(1, theta):
if (theta + e) % 2 == 0:
results.append((e * ksi, 1 * sgn(k_n)))
results.append((1 - e * ksi, -1 * sgn(k_n)))
# print("reversed")
for e in range(theta + 1, k + 1):
if (theta + e) % 2 != 0:
continue
results.append((e * ksi, -1 * sgn(k_n)))
results.append((1 - e * ksi, 1 * sgn(k_n)))
return SignatureFunction(values=results)
def get_signature_summand_as_theta_function(*arg):
def get_signture_function(theta):
# TBD: another formula (for t^2) description
k_n = abs(arg[-1])
if theta > k_n:
msg = "k for the pattern in the cable is " + str(arg[-1]) + \
". Parameter theta should not be larger than abs(k)."
raise ValueError(msg)
# twisted part
cable_signature = get_blanchfield_for_pattern(arg[-1], theta)
# untwisted part
for i, k in enumerate(arg[:-1][::-1]):
ksi = 1/(2 * k_n + 1)
power = 2^i
a = get_untwisted_signature_function(k)
shift = theta * ksi * power
b = a >> shift
c = a << shift
for _ in range(i):
b = b.double_cover()
c = c.double_cover()
cable_signature += b + c
test = b - c
test2 = -c + b
assert test == test
return cable_signature
get_signture_function.__doc__ = get_signture_function_docsting
return get_signture_function
def get_untwisted_signature_function(j):
# return the signature function of the T_{2,2k+1} torus knot
k = abs(j)
w = ([((2 * a + 1)/(4 * k + 2), -1 * sgn(j)) for a in range(k)] +
[((2 * a + 1)/(4 * k + 2), 1 * sgn(j))
for a in range(k + 1, 2 * k + 1)])
return SignatureFunction(values=w)
def extract_max(string):
numbers = re.findall('\d+', string)
numbers = map(int, numbers)
return max(numbers)
def mod_one(n):
return n - floor(n)
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)
"""
TorusCable.get_number_of_combinations_of_theta.__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)
"""
TorusCable.get_knot_descrption.__doc__ = \
"""
Arguments:
arbitrary number of lists of numbers, each list encodes a single cable.
Examples:
sage: get_knot_descrption([1, 3], [2], [-1, -2], [-3])
'T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)'
"""
mod_one.__doc__ = \
"""
Argument:
a number
Return:
the fractional part of the argument
Examples:
sage: mod_one(9 + 3/4)
3/4
sage: mod_one(-9 + 3/4)
3/4
sage: mod_one(-3/4)
1/4
"""
search_for_null_signature_value.__doc__ = \
"""
@ -367,6 +239,17 @@ search_for_null_signature_value.__doc__ = \
Data for knots that are candidates for slice knots are saved to a file.
"""
main.__doc__ = \
"""
This function is run if the script was called from the terminal.
It calls another function, search_for_null_signature_value,
to calculate signature functions for a schema
of a cable sum defined in the class Config.
Optionaly a parameter (a limit for k_0 value) can be given.
Thought to be run for time consuming calculations.
"""
extract_max.__doc__ = \
"""
Return:
@ -378,136 +261,6 @@ extract_max.__doc__ = \
3300
"""
TorusCable.eval_cable_for_null_signature.__doc__ = \
"""
This function calculates all possible twisted signature functions for
a knot that is given as an argument. The knot should be encoded as a list
of its direct component. Each component schould be presented as a list
of integers. This integers correspond to the k - values in each component/
cable. If a component is a mirror image of a cable the minus sign should
be written before each number for this component. For example:
eval_cable_for_null_signature([[1, 8], [2], [-2, -8], [-2]])
eval_cable_for_null_signature([[1, 2], [-1, -2]])
sage: eval_cable_for_null_signature([[1, 3], [2], [-1, -2], [-3]])
T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)
Zero cases: 1
All cases: 1225
Zero theta combinations:
(0, 0, 0, 0)
sage:
The numbers given to the function eval_cable_for_null_signature
are k-values for each component/cable in a direct sum.
"""
TorusCable.get_signature_as_function_of_theta.__doc__ = \
"""
Function intended to construct signature function for a connected
sum of multiple cables with varying theta parameter values.
Accept arbitrary number of arguments (depending on number of cables in
connected sum).
Each argument should be given as list of integer representing
k - parameters for a cable: parameters k_i (i=1,.., n-1) for satelit knots
T(2, 2k_i + 1) and - the last one - k_n for a pattern knot T(2, 2k_n + 1).
Returns a function that will take theta vector as an argument and return
an object SignatureFunction.
To calculate signature function for a cable sum and a theta values vector,
use as below.
sage: signature_function_generator = get_signature_as_function_of_theta(
[1, 3], [2], [-1, -2], [-3])
sage: sf = signature_function_generator(2, 1, 2, 2)
sage: print(sf)
0: 0
5/42: 1
1/7: 0
1/5: -1
7/30: -1
2/5: 1
3/7: 0
13/30: -1
19/42: -1
23/42: 1
17/30: 1
4/7: 0
3/5: -1
23/30: 1
4/5: 1
6/7: 0
37/42: -1
Or like below.
sage: print(get_signature_as_function_of_theta([1, 3], [2], [-1, -2], [-3]
)(2, 1, 2, 2))
0: 0
1/7: 0
1/6: 0
1/5: -1
2/5: 1
3/7: 0
1/2: 0
4/7: 0
3/5: -1
4/5: 1
5/6: 0
6/7: 0
"""
get_signature_summand_as_theta_function.__doc__ = \
"""
Argument:
n integers that encode a single cable, i.e.
values of q_i for T(2,q_0; 2,q_1; ... 2, q_n)
Return:
a function that returns SignatureFunction for this single cable
and a theta given as an argument
"""
SignatureFunction.__doc__ = \
"""
This simple class encodes twisted and untwisted signature functions
of knots. Since the signature function is entirely encoded by its signature
jump, the class stores only information about signature jumps
in a dictionary self.cnt_signature_jumps.
The dictionary stores data of the signature jump as a key/values pair,
where the key is the argument at which the functions jumps
and value encodes the value of the jump. Remember that we treat
signature functions as defined on the interval [0,1).
"""
get_signture_function_docsting = \
"""
This function returns SignatureFunction for previously defined single
cable T_(2, q) and a theta given as an argument.
The cable was defined by calling function
get_signature_summand_as_theta_function(*arg)
with the cable description as an argument.
It is an implementaion of the formula:
Bl_theta(K'_(2, d)) =
Bl_theta(T_2, d) + Bl(K')(ksi_l^(-theta) * t)
+ Bl(K')(ksi_l^theta * t)
"""
signature_as_function_of_theta_docstring = \
"""
Arguments:
Returns object of SignatureFunction class for a previously defined
connected sum of len(arg) cables.
Accept len(arg) arguments: for each cable one theta parameter.
If call with no arguments, all theta parameters are set to be 0.
"""
main.__doc__ = \
"""
This function is run if the script was called from the terminal.
It calls another function, search_for_null_signature_value,
to calculate signature functions for a schema
of a cable sum defined in the class Config.
Optionaly a parameter (a limit for k_0 value) can be given.
Thought to be run for time consuming calculations.
"""
if __name__ == '__main__':
global config