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first lines for cab.update function

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Maria Marchwicka 2020-09-24 02:42:25 +02:00
parent b729a4eeee
commit acbf651697
2 changed files with 90 additions and 33 deletions
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89
cable_signature.sage
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@ -7,6 +7,8 @@ from typing import Iterable
SIGNATURE = 0 SIGNATURE = 0
SIGMA = 1 SIGMA = 1
# 9.11 (9.8)
# 9.15 (9.9)
class SignatureFunction(object): class SignatureFunction(object):
@ -21,40 +23,57 @@ class SignatureFunction(object):
msg = "Signature function is defined on the interval [0, 1)." msg = "Signature function is defined on the interval [0, 1)."
assert all(k < 1 for k, v in values), msg assert all(k < 1 for k, v in values), msg
counter2 = collections.Counter({ k : v for k, v in values})
for k, v in values: for k, v in values:
counter[k] += v counter[k] += v
assert counter2 == counter
self.cnt_signature_jumps = counter self.cnt_signature_jumps = counter
self.tikz_plot("bum.tex")
def sum_of_absolute_values(self):
return sum([abs(i) for i in self.cnt_signature_jumps.values()])
def is_zero_everywhere(self): def is_zero_everywhere(self):
return not any(self.cnt_signature_jumps.values()) return not any(self.cnt_signature_jumps.values())
def double_cover(self): def double_cover(self):
# to read values for t^2 # to read values for t^2
items = self.cnt_signature_jumps.items()
counter = collections.Counter({ (1 + k) / 2 : v for k, v in items})
counter.update(collections.Counter({ k / 2 : v for k, v in items}))
new_data = [] new_data = []
for jump_arg, jump in self.cnt_signature_jumps.items(): for jump_arg, jump in self.cnt_signature_jumps.items():
if jump != 0: if jump != 0:
new_data.append((jump_arg/2, jump)) new_data.append((jump_arg/2, jump))
new_data.append((1/2 + jump_arg/2, jump)) new_data.append((1/2 + jump_arg/2, jump))
assert SignatureFunction(values=new_data) == SignatureFunction(counter=counter)
return SignatureFunction(values=new_data) return SignatureFunction(values=new_data)
def square_root(self): def square_root(self):
# to read values for t^(1/2) # to read values for t^(1/2)
new_data = [] new_data = []
for jump_arg, jump in self.cnt_signature_jumps.items(): for jump_arg, jump in self.cnt_signature_jumps.items():
if jump_arg < 1/2 and jump != 0: if jump_arg < 1/2:
new_data.append((2 * jump_arg, jump)) new_data.append((2 * jump_arg, jump))
counter = collections.Counter()
for jump_arg, jump in self.cnt_signature_jumps.items():
if jump_arg < 1/2:
counter[2 * jump_arg] = jump
assert SignatureFunction(values=new_data) == SignatureFunction(counter=counter)
return SignatureFunction(values=new_data) return SignatureFunction(values=new_data)
def minus_square_root(self): def minus_square_root(self):
# to read values for t^(1/2) # to read values for t^(1/2)
items = self.cnt_signature_jumps.items()
counter = collections.Counter() counter = collections.Counter()
for jump_arg, jump in self.cnt_signature_jumps.items(): for jump_arg, jump in self.cnt_signature_jumps.items():
if jump_arg >= 1/2 and jump != 0: if jump_arg >= 1/2:
counter[mod_one(2 * jump_arg)] = jump counter[mod_one(2 * jump_arg)] = jump
counter2 = collections.Counter({ mod_one(2 * k) : v for k, v in items if k >= 1/2 })
assert counter2 == counter
return SignatureFunction(counter=counter) return SignatureFunction(counter=counter)
def is_big(self): def is_big(self):
@ -63,12 +82,7 @@ class SignatureFunction(object):
items = sorted(self.cnt_signature_jumps.items()) items = sorted(self.cnt_signature_jumps.items())
for arg, jump in items: for arg, jump in items:
current += 2 * jump current += 2 * jump
msg = "current = " + str(current) + ", jump = " + str(jump) assert current == self(arg) + jump
msg += "\n" + str(self(arg))
result = [jump for jump_arg, jump in self.cnt_signature_jumps.items() if jump_arg < mod_one(arg)]
msg += "\nresult = " + str(sum(result))
msg += "\narg = " + str(arg)
assert current == self(arg) + jump, msg
if abs(current) > abs(max): if abs(current) > abs(max):
max = current max = current
# if abs(max) > 9: # if abs(max) > 9:
@ -81,6 +95,7 @@ class SignatureFunction(object):
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))
sf = SignatureFunction(values=new_data) sf = SignatureFunction(values=new_data)
counter = collections.Counter({mod_one(k + shift) : v \ counter = collections.Counter({mod_one(k + shift) : v \
for k,v in self.cnt_signature_jumps.items()}) for k,v in self.cnt_signature_jumps.items()})
assert SignatureFunction(counter=counter) == \ assert SignatureFunction(counter=counter) == \
@ -128,18 +143,29 @@ class SignatureFunction(object):
def total_sign_jump(self): def total_sign_jump(self):
# Total signature jump is the sum of all jumps. # Total signature jump is the sum of all jumps.
result = sum([j[1] for j in self.to_list()])
assert result == sum(v for _, v in self.cnt_signature_jumps.items())
return sum([j[1] for j in self.to_list()]) return sum([j[1] for j in self.to_list()])
def to_list(self): def to_list(self):
# Return signature jumps formated as a list # Return signature jumps formated as a list
assert sorted(self.cnt_signature_jumps.items(), key = lambda x: x[0]) == \
sorted(self.cnt_signature_jumps.items())
return sorted(self.cnt_signature_jumps.items(), key = lambda x: x[0]) return sorted(self.cnt_signature_jumps.items(), key = lambda x: x[0])
def step_function_data(self): def step_function_data(self):
# Transform the signature jump data to a format understandable # Transform the signature jump data to a format understandable
# by the plot function. # by the plot function.
l = self.to_list() l = self.to_list()
assert l == sorted(self.cnt_signature_jumps.items())
vals = ([(d[0], sum(2 * j[1] for j in l[:l.index(d)+1])) for d in l] + vals = ([(d[0], sum(2 * j[1] for j in l[:l.index(d)+1])) for d in l] +
[(0,self.cnt_signature_jumps[0]), (1,self.total_sign_jump())]) [(0,self.cnt_signature_jumps[0]), (1,self.total_sign_jump())])
print("step_function_data")
print(vals)
counter = copy(self.cnt_signature_jumps)
counter[0] = self.cnt_signature_jumps[0]
counter[1] = self.total_sign_jump()
print(sorted(counter.items()))
return vals return vals
def plot(self): def plot(self):
@ -147,7 +173,7 @@ class SignatureFunction(object):
plot_step_function(self.step_function_data()) plot_step_function(self.step_function_data())
def tikz_plot(self, file_name): def tikz_plot(self, file_name):
# Draw the graph of the signature and transform it into TiKz. # Draw the graph of the signature and transform it into TiKz.
# header of the LaTeX file # header of the LaTeX file
with open(file_name, "w") as output_file: with open(file_name, "w") as output_file:
@ -156,10 +182,12 @@ class SignatureFunction(object):
output_file.write("\\begin{document}\n") output_file.write("\\begin{document}\n")
output_file.write("\\begin{tikzpicture}\n") output_file.write("\\begin{tikzpicture}\n")
data = sorted(self.step_function_data()) data = sorted(self.step_function_data())
print("data")
print(data)
output_file.write(" \\datavisualization[scientific axes,visualize as smooth line,\n") output_file.write(" \\datavisualization[scientific axes,visualize as smooth line,\n")
output_file.write(" x axis={ticks={none,major={at={") output_file.write(" x axis={ticks={none,major={at={")
output_file.write(", " + str(N(data[0][0],digits=4)) + " as \\(" + str(data[0][0]) + "\\)") output_file.write(", " + str(N(data[0][0],digits=4)) + " as \\(" + str(data[0][0]) + "\\)")
for jump_arg,jump in data: for jump_arg, jump in data:
output_file.write(", " + str(N(jump_arg,digits=4)) + " as \\(" + str(jump_arg) + "\\)") output_file.write(", " + str(N(jump_arg,digits=4)) + " as \\(" + str(jump_arg) + "\\)")
output_file.write("}}}}\n") output_file.write("}}}}\n")
output_file.write(" ]\n") output_file.write(" ]\n")
@ -191,9 +219,28 @@ class TorusCable(object):
self.q_vector = q_vector self.q_vector = q_vector
k = k_vector k = k_vector
self.knot_sum = eval(knot_formula) self.knot_sum = eval(knot_formula)
self.knot_description = self.get_knot_descrption() self.knot_description = self.set_knot_descrption()
self.__sigma_function = None self.__sigma_function = None
# TBD property function
self.signature_as_function_of_theta = None self.signature_as_function_of_theta = None
self.signature_as_function_of_theta = self.get_signature_as_function_of_theta()
# self.signature_as_function_of_theta = None
def update(self, other):
# TBD knot_formula etc.
number_of_summands = len(self.knot_sum)
self.knot_description += " # " + other.knot_description
self.knot_formula = self.knot_formula[:-1] + ",\n" + \
other.knot_formula[1:]
print(self.knot_sum)
self.knot_sum += other.knot_sum
# self.signature_as_function_of_theta = \
# self.get_signature_as_function_of_theta() + \
# other.get_signature_as_function_of_theta()
print(self.knot_description)
print(self.knot_formula)
print(self.knot_sum)
def __get_sigma_function(self): def __get_sigma_function(self):
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]
@ -273,12 +320,18 @@ class TorusCable(object):
print("one vector " + str(thetas)) print("one vector " + str(thetas))
print("max sf " + str(sf.is_big())) print("max sf " + str(sf.is_big()))
print() print()
assert untwisted_part.is_zero_everywhere() # assert untwisted_part.is_zero_everywhere()
if verbose: if verbose:
print() print()
print(str(thetas)) print(str(thetas))
print(sf) print(sf)
msg = "tota signature jump = " + str(sf.total_sign_jump())
msg += "\nfunction\n" + str(sf)
assert sf.total_sign_jump() == 0, msg
return sf return sf
signature_as_function_of_theta.__doc__ =\ signature_as_function_of_theta.__doc__ =\
signature_as_function_of_theta_docstring signature_as_function_of_theta_docstring
@ -354,7 +407,7 @@ class TorusCable(object):
return SignatureFunction(values=w) return SignatureFunction(values=w)
def get_knot_descrption(self): def set_knot_descrption(self):
description = "" description = ""
for knot in self.knot_sum: for knot in self.knot_sum:
if knot[0] < 0: if knot[0] < 0:
@ -728,12 +781,12 @@ TorusCable.get_number_of_combinations_of_theta.__doc__ = \
and q_j is the last q parameter for the component (a single cable) and q_j is the last q parameter for the component (a single cable)
""" """
TorusCable.get_knot_descrption.__doc__ = \ TorusCable.set_knot_descrption.__doc__ = \
""" """
Arguments: Arguments:
arbitrary number of lists of numbers, each list encodes a single cable. arbitrary number of lists of numbers, each list encodes a single cable.
Examples: Examples:
sage: get_knot_descrption([1, 3], [2], [-1, -2], [-3]) sage: set_knot_descrption([1, 3], [2], [-1, -2], [-3])
'T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)' 'T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)'
""" """

34
my_signature.sage
View File

@ -20,8 +20,16 @@ class Config(object):
# knot_formula is a schema for knots which signature function # knot_formula is a schema for knots which signature function
# will be calculated # will be calculated
self.knot_formula = "[[k[0], k[1], k[3]], [-k[1], -k[3]], \ self.knot_formula = "[[k[0], k[1], k[3]], " + \
[k[2], k[3]], [-k[0], -k[2], -k[3]]]" "[-k[1], -k[3]], " + \
"[k[2], k[3]], " + \
"[-k[0], -k[2], -k[3]]]"
# self.knot_formula = "[[k[0], k[1], k[4]], [-k[1], -k[3]], \
# [k[2], k[3]], [-k[0], -k[2], -k[4]]]"
#
#
# self.knot_formula = "[[k[3]], [-k[3]], \ # self.knot_formula = "[[k[3]], [-k[3]], \
# [k[3]], [-k[3]] ]" # [k[3]], [-k[3]] ]"
@ -42,10 +50,10 @@ class Config(object):
self.start_shift = 0 self.start_shift = 0
self.verbose = True self.verbose = True
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
@ -102,7 +110,7 @@ def set_parameters(knot_formula, limit, verbose, print_results):
if knot_formula is None: if knot_formula is None:
knot_formula = config.knot_formula knot_formula = config.knot_formula
if verbose is None: if verbose is None:
vebose = config.verbose verbose = 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 return knot_formula, limit, verbose, print_results
@ -127,9 +135,8 @@ def search_for_large_sigma_value(knot_formula=None, limit=None,
for c in combinations: for c in combinations:
q = [P.unrank(i + config.start_shift) for i in c] q = [P.unrank(i + config.start_shift) for i in c]
if config.only_slice_candidates: if config.only_slice_candidates:
if not (q[3] > 4 * q[2] and ratio = q[3] > 4 * q[2] and q[2] > 4 * q[1] and q[1] > 4 * q[0]
q[2] > 4 * q[1] and if not ratio:
q[1] > 4 * q[0]):
if verbose: if verbose:
print("Ratio-condition does not hold") print("Ratio-condition does not hold")
continue continue
@ -143,9 +150,6 @@ def search_for_large_sigma_value(knot_formula=None, limit=None,
# 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,
verbose=None, print_results=None): verbose=None, print_results=None):
@ -154,6 +158,8 @@ def search_for_null_signature_value(knot_formula=None, limit=None,
set_parameters(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)
combinations = it.combinations_with_replacement(range(1, limit + 1), combinations = it.combinations_with_replacement(range(1, limit + 1),
k_vector_size) k_vector_size)
with open(config.f_results, 'w') as f_results: with open(config.f_results, 'w') as f_results:
@ -202,11 +208,9 @@ 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:
q = [P.unrank(i + config.start_shift) for i in c] q = [P.unrank(i + config.start_shift) for i in c]
q[3] = 79
if config.only_slice_candidates: if config.only_slice_candidates:
if not (q[3] > 4 * q[2] and ratio = q[3] > 4 * q[2] and q[2] > 4 * q[1] and q[1] > 4 * q[0]
q[2] > 4 * q[1] and if not ratio:
q[1] > 4 * q[0]):
if verbose: if verbose:
print("Ratio-condition does not hold") print("Ratio-condition does not hold")
continue continue