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docstings for few functions

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Maria Marchwicka 2020-07-10 02:00:12 +02:00
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my_signature.sage
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@ -1,13 +1,48 @@
#!/usr/bin/python
# TBD: remove part of the description to readme/example
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):
form = calculate_form(v, v, q4)
if form:
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).
The script can be run as a sage script from the terminal or used in interactive
modeselfself.
The script can be run as a sage script from the terminal
or used in interactive mode.
A knot (cable sum) is encoded as a list where each element (also a list)
corresponds to a cable knot.
corresponds to a cable knot, e.g. a list
[[1, 3], [2], [-1, -2], [-3]] encodes
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.
@ -26,48 +61,8 @@ sage:
The numbers given to the function eval_cable_for_thetas 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 as follow:
sage: signature_function_generator = get_function_of_theta_for_sum([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
sage:
or like below:
sage: print get_function_of_theta_for_sum([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
sage:
get_function_of_theta_for_sum (see help/docstring for details).
"""
import os
@ -80,7 +75,6 @@ import pandas as pd
import re
class MySettings(object):
def __init__(self):
self.f_results = os.path.join(os.getcwd(), "results.out")
@ -94,6 +88,19 @@ class MySettings(object):
# 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_sum_formula = "[[k[0], k[1], k[2]], [k[3]],\
# [-k[0], -k[1], -k[3]], [-k[2]]]"
self.default_limit = 3
@ -119,7 +126,6 @@ class SignatureFunction(object):
"Signature function is defined on the interval [0, 1)."
self.data[jump_arg] = jump
def sum_of_absolute_values(self):
return sum([abs(i) for i in self.data.values()])
@ -142,7 +148,6 @@ class SignatureFunction(object):
def get_signture_jump(self, t):
return self.data.get(t, 0)
def minus_square_root(self):
# to read values for t^(1/2)
new_data = []
@ -151,7 +156,6 @@ class SignatureFunction(object):
new_data.append((mod_one(2 * jump_arg), jump))
return SignatureFunction(new_data)
def __lshift__(self, shift):
# A shift of the signature functions corresponds to the rotation.
return self.__rshift__(-shift)
@ -187,19 +191,16 @@ class SignatureFunction(object):
return ''.join([str(jump_arg) + ": " + str(jump) + "\n"
for jump_arg, jump in sorted(self.data.items())])
def get_untwisted_signature():
return 0
def main(arg):
"""
This function is run if the script was called from the terminal.
It calls another function to calculate signature functions for a schema
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:
@ -212,10 +213,11 @@ 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 to the stdout.
limit is the upper bound for the first value in k_vector, i.e first k value
in a cable sum (the number of knots that will be constracted depends
on limit value).
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
@ -313,14 +315,18 @@ def get_blanchfield_for_pattern(k_n, theta):
def get_cable_signature_as_theta_function(*arg):
"""
This function takes as an argument a single cable and returns another
function that alow to calculate signature function for previously defined
cable and a theta given as an argument.
This function takes as an argument a single cable T_(2, q), i.e.
arbitrary number of integers that encode the cable,
and returns another function that alow to calculate signature function
for this single cable and a theta given as an argument.
"""
def get_signture_function(theta):
"""
This function returns SignatureFunction for previously defined cable
and a theta given as an argument.
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)
@ -362,7 +368,7 @@ def get_untwisted_signature_function(j):
def get_function_of_theta_for_sum(*arg):
"""
Function intended to calculate signature function for a connected
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).
@ -371,6 +377,47 @@ def get_function_of_theta_for_sum(*arg):
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_function_of_theta_for_sum(
[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_function_of_theta_for_sum([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
"""
def signature_function_for_sum(*thetas, **kwargs):
@ -387,16 +434,23 @@ def get_function_of_theta_for_sum(*arg):
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."
@ -419,6 +473,20 @@ def eval_cable_for_thetas(knot_sum, print_results=True, verbose=False):
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)
@ -453,18 +521,6 @@ def eval_cable_for_thetas(knot_sum, print_results=True, verbose=False):
return None
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]
def check_squares(a, k):
print
p = 2 * k + 1
@ -483,6 +539,16 @@ def check_squares(a, k):
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)
@ -490,16 +556,56 @@ def get_number_of_combinations(*arg):
def extract_max(string):
"""This function returns maximal number from given 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):
"""This function returns the fractional part of some number."""
"""
Argument:
a number
Return:
the fractional part of a number
Examples:
sage: mod_one(9 + 3/4)
3/4
sage: mod_one(-9 + 3/4)
3/4
sage: mod_one(-3/4)
1/4
"""
return n - floor(n)
def get_knot_descrption(*arg):
"""
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)'
"""
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]
if __name__ == '__main__' and '__file__' in globals():
# not called in interactive mode as __file__ is not defined
main(sys.argv)