classes as separate module - now working
This commit is contained in:
Maria Marchwicka
parent
0110e6161c
commit
d9986b63c1
@ -1,6 +1,8 @@
|
|||||||
#!/usr/bin/python
|
#!/usr/bin/python
|
||||||
import collections
|
import collections
|
||||||
|
import numpy as np
|
||||||
|
import itertools as it
|
||||||
|
import re
|
||||||
|
|
||||||
class TorusCable(object):
|
class TorusCable(object):
|
||||||
def __init__(self, knot_formula, k_vector=None, q_vector=None):
|
def __init__(self, knot_formula, k_vector=None, q_vector=None):
|
||||||
@ -514,3 +516,290 @@ class SignatureFunction(object):
|
|||||||
|
|
||||||
def mod_one(n):
|
def mod_one(n):
|
||||||
return n - floor(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
|
||||||
|
"""
|
||||||
|
|||||||
@ -11,6 +11,11 @@ import collections
|
|||||||
import itertools as it
|
import itertools as it
|
||||||
import numpy as np
|
import numpy as np
|
||||||
import re
|
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
|
from cable_signature import SignatureFunction, TorusCable
|
||||||
|
|
||||||
class Config(object):
|
class Config(object):
|
||||||
@ -201,6 +206,11 @@ 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)
|
||||||
|
|
||||||
|
def extract_max(string):
|
||||||
|
numbers = re.findall('\d+', string)
|
||||||
|
numbers = map(int, 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:
|
||||||
@ -209,144 +219,6 @@ def is_trivial_combination(knot_sum):
|
|||||||
return True
|
return True
|
||||||
return False
|
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__ = \
|
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.
|
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__ = \
|
extract_max.__doc__ = \
|
||||||
"""
|
"""
|
||||||
Return:
|
Return:
|
||||||
@ -378,136 +261,6 @@ extract_max.__doc__ = \
|
|||||||
3300
|
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__':
|
if __name__ == '__main__':
|
||||||
global config
|
global config
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user