script from Wojtek and my first version
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Maria Marchwicka
parent
f64fe6f905
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7b3f3f8a23
136
my_signature.sage
Normal file
136
my_signature.sage
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#!/usr/bin/env python
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import collections
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def mod_one(n):
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"""This function returns the fractional part of some number."""
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if n >= 1:
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return mod_one(n - 1)
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if n < 0:
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return mod_one(n + 1)
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return n
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class av_signature_function(object):
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'''
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This simple class encodes twisted and untwisted signature functions
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of knots. Since the signature function is entirely encoded by its signature
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jump, the class stores only information about signature jumps
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in a dictionary self.data.
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The dictionary stores data of the signature jump as a key/values pair,
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where the key is the argument at which the functions jumps
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and value encodes the value of the jump. Remember that we treat
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signature functions as defined on the interval [0,1).
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'''
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def __init__(self, values=[]):
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# We will store data of signature jumps here.
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self.data = collections.defaultdict(int)
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# values contain initial data of singature jumps
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for jump_arg, jump in values:
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assert 0 <= jump_arg < 1, \
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"Signature function is defined on the interval [0, 1)."
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################################### what for += ???
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self.data[jump_arg] = jump
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def value(self, arg):
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# Compute the value of the signature function at the point arg.
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# This requires summing all signature jumps that occur before arg.
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assert 0 <= arg < 1, \
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"Signature function is defined on the interval [0, 1)."
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val = 0
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for jump_arg, jump in self.data.items():
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if jump_arg < arg:
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val += 2 * jump
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elif jump_arg == arg:
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val += jump
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return val
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############## what for - it is == 0
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def total_sign_jump(self):
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# Total signature jump is the sum of all jumps.
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a = sum([j[1] for j in self.to_list()])
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b = sum(self.data.values())
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# print b
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assert a == b
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return sum(self.data.values())
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def to_list(self):
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# Return signature jumps formated as a list
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return sorted(self.data.items(), key=lambda x: x[0])
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def step_function_data(self):
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# Transform the signature jump data to a format understandable
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# by the plot function.
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l = self.to_list()
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vals = ([(d[0], sum(2 * j[1] for j in l[:l.index(d)+1])) for d in l] +
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[(0, self.data[0]), (1, self.total_sign_jump())])
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return vals
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def plot(self):
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# plot the signture function
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plot_step_function(self.step_function_data())
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def tikz_plot(self, file_name):
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# Draw the graph of the signature and transform it into TiKz.
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# header of the LaTeX file
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output_file = open(file_name, "w")
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output_file.write("\\documentclass[tikz]{standalone}\n")
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output_file.write("\\usetikzlibrary{datavisualization,datavisualization.formats.functions}\n")
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output_file.write("\\begin{document}\n")
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output_file.write("\\begin{tikzpicture}\n")
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data = sorted(self.step_function_data())
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output_file.write(" \\datavisualization[scientific axes,visualize as smooth line,\n")
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output_file.write(" x axis={ticks={none,major={at={")
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output_file.write(", " + str(N(data[0][0], digits=4)) + " as \\(" + str(data[0][0]) + "\\)")
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for jump_arg, jump in data[1:]:
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output_file.write(", " + str(N(jump_arg, digits=4)) + " as \\(" + str(jump_arg) + "\\)")
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output_file.write("}}}}\n")
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output_file.write(" ]\n")
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output_file.write("data [format=function]{\n")
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output_file.write("var x : interval [0:1];\n")
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output_file.write("func y = \\value x;\n")
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output_file.write("};\n")
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# close LaTeX enviroments
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output_file.write("\\end{tikzpicture}\n")
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output_file.write("\\end{document}\n")
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output_file.close()
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def __lshift__(self, shift):
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# Shift of the signature functions correspond to the rotations.
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return self.__rshift__(-shift)
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def __rshift__(self, shift):
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new_data = []
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for jump_arg, jump in self.data.items():
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new_data.append((mod_one(jump_arg + shift), jump))
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return av_signature_function(new_data)
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def __sub__(self, other):
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# we cn perform arithmetic operations on signature functions.
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return self + other.__neg__()
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def __neg__(self):
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for jump_arg in self.data.keys():
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self.data[jump_arg] *= -1
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return self
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def __add__(self, other):
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for jump_arg, jump in other.data.items():
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self.data[jump_arg] += jump
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return self
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def __str__(self):
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return '\n'.join([str(jump_arg) + ": " + str(jump)
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for jump_arg, jump in self.data.items()])
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def __repr__(self):
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return self.__str__()
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def untw_signature(k):
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# Return the signature function of the T_{2,2k+1} torus knot.
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l = ([((2 * a + 1)/(4 * k + 2), -1) for a in range(k)] +
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[((2 * a + 1)/(4 * k + 2), 1) for a in range(k + 1, 2 * k + 1)])
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# print l
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# print type(l)
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return av_signature_function(l)
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125
signature.sage
Normal file
125
signature.sage
Normal file
@ -0,0 +1,125 @@
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#!/usr/bin/env python
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import collections
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def mod_one(n):
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"""This function returns the fractional part of some number."""
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if n >= 1:
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return mod_one(n - 1)
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if n < 0:
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return mod_one(n + 1)
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else:
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return n
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class av_signature_function(object):
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'''
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This simple class encodes twisted and untwisted signature functions
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of knots. Since the signature function is entirely encoded by its signature
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jump, the class stores only information about signature jumps
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in a dictionary self.data.
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The dictionary stores data of the signature jump as a key/values pair,
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where the key is the argument at which the functions jumps
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and value encodes the value of the jump.
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Remember that we treat signature functions as defined on the interval [0,1).
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'''
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def __init__(self,values=[]):
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# We will store data of signature jumps here.
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self.data = collections.defaultdict(int)
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# values contain initial data of singature jumps
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for jump_arg, jump in values:
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assert 0 <= jump_arg and jump_arg < 1
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self.data[jump_arg] += jump
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def value(self, arg):
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# Compute the value of the signature function at the point arg.
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# This requires summing all signature jumps that occur before arg.
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assert 0 <= arg and arg < 1
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val = 0
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for jump_arg, jump in self.data.items():
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if jump_arg < arg:
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val += 2 * jump
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elif jump_arg == arg:
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val += jump
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return val
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def total_sign_jump(self):
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# Total signature jump is the sum of all jumps.
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return sum([j[1] for j in self.to_list()])
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def to_list(self):
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# Return signature jumps formated as a list
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return sorted(self.data.items(), key = lambda x: x[0])
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def step_function_data(self):
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# Transform the signature jump data to a format understandable
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# by the plot function.
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l = self.to_list()
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vals = ([(d[0], sum(2 * j[1] for j in l[:l.index(d)+1])) for d in l] +
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[(0,self.data[0]), (1,self.total_sign_jump())])
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return vals
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def plot(self):
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# plot the signture function
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plot_step_function(self.step_function_data())
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def tikz_plot(self, file_name):
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# Draw the graph of the signature and transform it into TiKz.
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# header of the LaTeX file
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output_file = open(file_name, "w")
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output_file.write("\\documentclass[tikz]{standalone}\n")
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output_file.write("\\usetikzlibrary{datavisualization,datavisualization.formats.functions}\n")
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output_file.write("\\begin{document}\n")
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output_file.write("\\begin{tikzpicture}\n")
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data = sorted(self.step_function_data())
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output_file.write(" \\datavisualization[scientific axes,visualize as smooth line,\n")
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output_file.write(" x axis={ticks={none,major={at={")
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output_file.write(", " + str(N(data[0][0],digits=4)) + " as \\(" + str(data[0][0]) + "\\)")
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for jump_arg,jump in data[1:]:
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output_file.write(", " + str(N(jump_arg,digits=4)) + " as \\(" + str(jump_arg) + "\\)")
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output_file.write("}}}}\n")
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output_file.write(" ]\n")
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output_file.write("data [format=function]{\n")
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output_file.write("var x : interval [0:1];\n")
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output_file.write("func y = \\value x;\n")
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output_file.write("};\n")
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# close LaTeX enviroments
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output_file.write("\\end{tikzpicture}\n")
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output_file.write("\\end{document}\n")
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output_file.close()
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def __lshift__(self, shift):
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# Shift of the signture functions correspond to the rotations.
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return self.__rshift__(-shift)
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def __rshift__(self, shift):
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new_data = list()
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for jump_arg, jump in self.data.items():
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new_data.append((mod_one(jump_arg + shift),jump))
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return av_signature_function(new_data)
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def __sub__(self, other):
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# we cn perform arithmetic operations on signature functions.
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return self + other.__neg__()
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def __neg__(self):
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for jump_arg in self.data.keys():
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self.data[jump_arg] *= (-1)
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return self
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def __add__(self, other):
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for jump_arg, jump in other.data.items():
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self.data[jump_arg] += jump
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return self
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def __str__(self):
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return '\n'.join([str(jump_arg) + ": " + str(jump)
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for jump_arg, jump in self.data.items()])
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def __repr__(self):
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print self.__str__()
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def untw_signature(k):
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# Return the signture function of the T_{2,2k+1} torus knot.
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l = ([((2 * a + 1)/(4 * k + 2), -1) for a in range(k)] +
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[((2 * a + 1)/(4 * k + 2), 1) for a in range(k + 1, 2 * k + 1)])
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return av_signature_function(l)
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