766 lines
25 KiB
Python
766 lines
25 KiB
Python
#!/usr/bin/python
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import numpy as np
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import itertools as it
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from typing import Iterable
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from collections import Counter
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from sage.arith.functions import LCM_list
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import warnings
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import re
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import matplotlib.pyplot as plt
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import inspect
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# 9.11 (9.8)
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# 9.15 (9.9)
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class SignatureFunction():
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def __init__(self, values=None, counter=None):
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# counter of signature jumps
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if counter is None:
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counter = Counter()
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if values is None:
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values = []
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for k, v in values:
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counter[k] += v
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counter = Counter({k : v for k, v in counter.items() if v != 0})
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if any(k >= 1 for k in counter.keys()):
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msg = "Signature function is defined on the interval [0, 1)."
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raise ValueError(msg)
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counter[0] += 0
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counter[1] += 0
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self.jumps_counter = counter
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def __rshift__(self, shift):
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# A shift of the signature functions corresponds to the rotation.
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counter = Counter({mod_one(k + shift) : v \
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for k, v in self.jumps_counter.items()})
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return SignatureFunction(counter=counter)
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def __lshift__(self, shift):
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return self.__rshift__(-shift)
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def __neg__(self):
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counter = Counter()
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counter.subtract(self.jumps_counter)
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return SignatureFunction(counter=counter)
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def __add__(self, other):
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counter = copy(self.jumps_counter)
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counter.update(other.jumps_counter)
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return SignatureFunction(counter=counter)
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def __sub__(self, other):
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counter = copy(self.jumps_counter)
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counter.subtract(other.jumps_counter)
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return SignatureFunction(counter=counter)
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def __eq__(self, other):
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return self.jumps_counter == other.jumps_counter
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def __str__(self):
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result = ''.join([str(jump_arg) + ": " + str(jump) + "\n"
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for jump_arg, jump in sorted(self.jumps_counter.items())])
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return result
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def __repr__(self):
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result = ''.join([str(jump_arg) + ": " + str(jump) + ", "
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for jump_arg, jump in sorted(self.jumps_counter.items())])
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return result[:-2] + "."
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def __call__(self, arg):
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# return the value of the signature function at the point arg, i.e.
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# sum of all signature jumps that occur before arg
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items = self.jumps_counter.items()
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result = [jump for jump_arg, jump in items if jump_arg < mod_one(arg)]
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return 2 * sum(result) + self.jumps_counter[arg]
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def is_zero_everywhere(self):
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return not any(self.jumps_counter.values())
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def double_cover(self):
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# to read values for t^2
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items = self.jumps_counter.items()
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counter = Counter({(1 + k) / 2 : v for k, v in items})
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counter.update(Counter({k / 2 : v for k, v in items}))
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return SignatureFunction(counter=counter)
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def square_root(self):
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# to read values for t^(1/2)
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counter = Counter()
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for jump_arg, jump in self.jumps_counter.items():
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if jump_arg < 1/2:
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counter[2 * jump_arg] = jump
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return SignatureFunction(counter=counter)
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def minus_square_root(self):
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# to read values for t^(1/2)
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items = self.jumps_counter.items()
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counter = Counter({mod_one(2 * k) : v for k, v in items if k >= 1/2})
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return SignatureFunction(counter=counter)
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def extremum(self, limit=None):
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max = 0
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current = 0
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items = sorted(self.jumps_counter.items())
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for arg, jump in items:
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current += 2 * jump
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assert current == self(arg) + jump
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if abs(current) > abs(max):
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max = current
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if limit is not None:
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if abs(max) > limit:
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break
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return max
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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 sorted(self.jumps_counter.items())])
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class SignatureWriter():
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def __init__(self, signature_function):
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self.sf = signature_function
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def plot(self, title=None, subplot=False):
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keys = sorted(self.sf.jumps_counter.keys())
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y = [self.sf(k) + self.sf.jumps_counter[k] for k in keys]
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xmax = [k for k in keys if k != 0]
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xmin = [k for k in keys if k != 1]
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fig, ax = plt.subplots(1, 1)
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ax.set(ylabel='signature function')
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if title is not None:
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ax.set(title=title)
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ax.hlines(y, xmin, xmax, color='blue')
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plt.savefig('sf.png')
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plt.close()
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from PIL import Image
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image = Image.open('sf.png')
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image.show()
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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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result = [(k, self.sf(k) + self.sf.jumps_counter[k])
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for k in sorted(self.sf.jumps_counter.keys())]
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return result
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def tikz_plot(self, file_name):
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plt_sin = plot(sin(x), (x, 0, 2*pi))
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# plt_sin.show()
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plt_sin.save("MyPic.pdf")
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return
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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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head = inspect.cleandoc(
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r"""
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\documentclass{standalone}
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\usepackage{tikz}
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\usetikzlibrary{calc}
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\begin{document}
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\begin{tikzpicture}
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""")
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body = \
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r"""
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%A piecewise linear function is drawn over the interval.
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\draw (5,0) -- (6,-4);
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%The axes are drawn.
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\draw[latex-latex] ($(0,{-4*(2/5)}) +(0pt,-12.5pt)$) --
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($(0,{4*(2/5)}) +(0pt,12.5pt)$) node[above right]{$y$};
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\draw[latex-latex] ($({-4*(2/5)},0) +(-12.5pt,0pt)$) --
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($({12*(2/5)},0) +(12.5pt,0pt)$) node[below right]{$x$};
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"""
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tail = \
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r"""
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\end{tikzpicture}
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\end{document}
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"""
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tikzpicture = re.sub(r' +', ' ', ''.join([head, body, tail]))
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tikzpicture = re.sub(r'\n ', '\n', tikzpicture)
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with open("tmp.tex", "w") as f:
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f.write(tikzpicture)
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data = self.step_function_data()
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with open(file_name, "w") as f:
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head = \
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r"""
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\documentclass[tikz]{{standalone}}
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%\usepackage{{tikz}}
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\usetikzlibrary{{datavisualization}}
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\usetikzlibrary{{datavisualization.formats.functions}}
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%\usetikzlibrary{{calc}}
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\begin{{document}}
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\begin{{tikzpicture}}
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\datavisualization[scientific axes, visualize as smooth line,
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x axis={{ticks={{none,major={{at={{, {arg0} " as \\( {val0} \\
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%]
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""".format(arg0=str(N(data[0][0] ,digits=4)), val0=str(data[0][0]))
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f.write(head)
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# f.write(", " + str(N(data[0][0],digits=4)) + " as \\(" + \
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# str(data[0][0]) + "\\)")
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for jump_arg, jump in data[1:3]:
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f.write(", " + str(N(jump_arg,digits=4)) +
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" as \\(" + str(jump_arg) + "\\)")
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f.write("}}}}\n")
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f.write(" ]\n")
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f.write("data [format=function]{\n")
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f.write("var x : interval [0:1];\n")
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f.write("func y = \\value x;\n")
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f.write("};\n")
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# close LaTeX enviroments
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tail = \
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r"""
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%};
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\end{tikzpicture}
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\end{document}
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"""
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f.write(tail)
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class CableSummand():
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pass
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class CableSum():
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def __init__(self, knot_formula, k_vector=None, q_vector=None):
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self._knot_formula = knot_formula
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# q_i = 2 * k_i + 1
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if k_vector is not None:
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self.k_vector = k_vector
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elif q_vector is not None:
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self.q_vector = q_vector
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else:
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self.q_vector = self.get_q_vector_alg_slice(self.knot_formula)
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self._signature_as_function_of_theta = None
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@property
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def signature_as_function_of_theta(self):
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if self._signature_as_function_of_theta is None:
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self._signature_as_function_of_theta = \
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self.get_signature_as_function_of_theta()
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return self._signature_as_function_of_theta
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# knot encoding
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@property
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def knot_formula(self):
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return self._knot_formula
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@property
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def knot_description(self):
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return self._knot_description
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# knot encoding
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@property
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def knot_sum(self):
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return self._knot_sum
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@knot_sum.setter
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def knot_sum(self, knot_sum):
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self._knot_sum = knot_sum
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self._knot_description = self.get_knot_descrption(knot_sum)
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self._patt_k_list = [abs(i[-1]) for i in knot_sum]
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self._patt_q_list = [2 * i + 1 for i in self._patt_k_list]
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if any(n not in Primes() for n in self._patt_q_list):
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msg = "Incorrect q-vector. This implementation assumes that" + \
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" all last q values are prime numbers.\n" + \
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str(self._patt_q_list)
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raise ValueError(msg)
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self.q_order = LCM_list(self._patt_q_list)
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@property
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def patt_k_list(self):
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return self._patt_k_list
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@property
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def patt_q_list(self):
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return self._patt_q_list
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# q_order is LCM of all q values for pattern knots
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@property
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def q_order(self):
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return self._q_order
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@q_order.setter
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def q_order(self, val):
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self._q_order = val
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@property
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def k_vector(self):
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return self._k_vector
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@k_vector.setter
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def k_vector(self, k):
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self._k_vector = k
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if self.extract_max(self.knot_formula) > len(k) - 1:
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msg = "The vector for knot_formula evaluation is to short!"
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msg += "\nk_vector " + str(k) + " \nknot_formula " \
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+ str(self.knot_formula)
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raise IndexError(msg)
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self.knot_sum = eval(self.knot_formula)
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self._q_vector = [2 * k_val + 1 for k_val in k]
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@property
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def q_vector(self):
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return self._q_vector
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@q_vector.setter
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def q_vector(self, new_q_vector):
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self.k_vector = [(q - 1)/2 for q in new_q_vector]
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def __add__(self, other):
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s_formula = self.knot_formula
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o_formula = other.knot_formula
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k_vector = self.k_vector
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if self.k_vector != other.k_vector:
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msg = "k_vectors are different. k-vector preserving addition is " +\
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"impossible."
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warnings.warn(msg)
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shift = len(self.k_vector)
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o_formula = re.sub(r'\d+', lambda x: str(int(x.group()) + shift),
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o_formula)
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k_vector += other.k_vector
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knot_formula = s_formula[:-1] + ",\n" + o_formula[1:]
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cable = CableSum(knot_formula, k_vector=k_vector)
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s_sig = self.signature_as_function_of_theta
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o_sig = other.signature_as_function_of_theta
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shift = len(self.knot_sum)
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def signature_as_function_of_theta(*thetas, **kwargs):
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thetas = cable.parse_thetas(*thetas)
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result = s_sig(*thetas[shift:]) + o_sig(*thetas[0:shift])
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return result
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cable._signature_as_function_of_theta = signature_as_function_of_theta
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return cable
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def parse_thetas(self, *thetas):
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summands_num = len(self.knot_sum)
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if not thetas:
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return summands_num * (0,)
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if len(thetas) == 1 and summands_num > 1:
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if isinstance(thetas[0], Iterable):
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if len(thetas[0]) >= summands_num:
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return tuple(thetas[0])
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elif not thetas[0]:
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return summands_num * (0,)
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elif thetas[0] == 0:
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return summands_num * (0,)
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else:
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msg = "This function takes at least " + str(summands_num) + \
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" arguments or no argument at all (" + str(len(thetas)) \
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+ " given)."
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raise TypeError(msg)
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return tuple(thetas)
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@staticmethod
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def get_q_vector_alg_slice(knot_formula):
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lowest_number = 2
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q_vector = [0] * (CableSum.extract_max(knot_formula) + 1)
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P = Primes()
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for layer in CableSum.get_layers_from_formula(knot_formula)[::-1]:
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for el in layer:
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q_vector[el] = P.next(lowest_number)
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lowest_number = q_vector[el]
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lowest_number *= 4
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return q_vector
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@staticmethod
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def extract_max(string):
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numbers = re.findall(r'\d+', string)
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numbers = map(int, numbers)
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return max(numbers)
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@staticmethod
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def get_blanchfield_for_pattern(k_n, theta=0):
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msg = "Theorem on which this function is based, assumes " +\
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"theta < k, where q = 2*k + 1 for pattern knot T(p, q)."
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if theta == 0:
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sf = CableSum.get_untwisted_signature_function(k_n)
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return sf.square_root() + sf.minus_square_root()
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k = abs(k_n)
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assert theta <= k, msg
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results = []
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ksi = 1/(2 * k + 1)
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counter = Counter()
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# print("lambda_odd, i.e. (theta + e) % 2 != 0")
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for e in range(1, k + 1):
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if (theta + e) % 2 != 0:
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counter[e * ksi] = 1 * sgn(k_n)
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counter[1 - e * ksi] = -1 * sgn(k_n)
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results.append((e * ksi, 1 * sgn(k_n)))
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results.append((1 - e * ksi, -1 * sgn(k_n)))
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# for example for k = 9 (q = 19) from this part we get
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# for even theta
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# 2/19: 1
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# 4/19: 1
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# 6/19: 1
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# 8/19: 1
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# 11/19: -1
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# 13/19: -1
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# 15/19: -1
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# 17/19: -1
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#
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# for odd theta
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# 1/19: 1
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# 3/19: 1
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# 5/19: 1
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# 7/19: 1
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# 9/19: 1
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# 10/19: -1
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# 12/19: -1
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# 14/19: -1
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# 16/19: -1
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# 18/19: -1
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# print("lambda_even")
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# print("normal")
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for e in range(1, theta):
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if (theta + e) % 2 == 0:
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results.append((e * ksi, 1 * sgn(k_n)))
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results.append((1 - e * ksi, -1 * sgn(k_n)))
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# print("reversed")
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for e in range(theta + 1, k + 1):
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if (theta + e) % 2 == 0:
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results.append((e * ksi, -1 * sgn(k_n)))
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results.append((1 - e * ksi, 1 * sgn(k_n)))
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return SignatureFunction(values=results)
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@staticmethod
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def get_untwisted_signature_function(j):
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# return the signature function of the T_{2, 2k+1} torus knot
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k = abs(j)
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q = 2 * k + 1
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counter = Counter({(2 * a + 1)/(2 * q) : -sgn(j)
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for a in range(k)})
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counter.update(Counter({(2 * a + 1)/(2 * q) : sgn(j)
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for a in range(k + 1, q)}))
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return SignatureFunction(counter=counter)
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@staticmethod
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def get_knot_descrption(knot_sum):
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description = ""
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for knot in knot_sum:
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if knot[0] < 0:
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description += "-"
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description += "T("
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for k in knot:
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description += "2, " + str(2 * abs(k) + 1) + "; "
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description = description[:-2] + ") # "
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return description[:-3]
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@staticmethod
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def get_layers_from_formula(knot_formula):
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k_indices = re.sub(r'[k-]', '', knot_formula)
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k_indices = re.sub(r'\[\d+\]', lambda x: x.group()[1:-1], k_indices)
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k_indices = eval(k_indices)
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number_of_layers = max(len(lst) for lst in k_indices)
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layers = []
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for i in range(1, number_of_layers + 1):
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layer = set()
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for lst in k_indices:
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if len(lst) >= i:
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layer.add(lst[-i])
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layers.append(layer)
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return layers
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def get_signature_as_function_of_theta(self, **key_args):
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if 'verbose' in key_args:
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verbose_default = key_args['verbose']
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else:
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verbose_default = False
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knot_desc = self.knot_description
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def signature_as_function_of_theta(*thetas, **kwargs):
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# print("\n\nsignature_as_function_of_theta " + knot_desc)
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verbose = verbose_default
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if 'verbose' in kwargs:
|
|
verbose = kwargs['verbose']
|
|
thetas = self.parse_thetas(*thetas)
|
|
|
|
untwisted_part = SignatureFunction()
|
|
twisted_part = SignatureFunction()
|
|
|
|
# for each cable knot (summand) in cable sum apply theta
|
|
for i, knot in enumerate(self.knot_sum):
|
|
ssf = self.get_summand_signature_as_theta_function(*knot)
|
|
tp, up = ssf(thetas[i])
|
|
twisted_part += tp
|
|
untwisted_part += up
|
|
sf = twisted_part + untwisted_part
|
|
|
|
if verbose:
|
|
print()
|
|
print(str(thetas))
|
|
print(sf)
|
|
assert sf.total_sign_jump() == 0
|
|
return sf
|
|
|
|
signature_as_function_of_theta.__doc__ =\
|
|
signature_as_function_of_theta_docstring
|
|
return signature_as_function_of_theta
|
|
|
|
def get_untwisted_part(self, *knot_as_k_values, theta=0):
|
|
patt_k = knot_as_k_values[-1]
|
|
ksi = 1/(2 * abs(patt_k) + 1)
|
|
|
|
untwisted_part = SignatureFunction()
|
|
# For each knot summand consider k values in reversed order,
|
|
# ommit k value for pattern.
|
|
for layer_num, k in enumerate(knot_as_k_values[:-1][::-1]):
|
|
sf = CableSum.get_untwisted_signature_function(k)
|
|
shift = theta * ksi * 2^layer_num
|
|
right_shift = sf >> shift
|
|
left__shift = sf << shift
|
|
for _ in range(layer_num):
|
|
right_shift = right_shift.double_cover()
|
|
left__shift = left__shift.double_cover()
|
|
untwisted_part += right_shift + left__shift
|
|
return untwisted_part
|
|
|
|
def get_summand_signature_as_theta_function(self, *knot_as_k_values):
|
|
|
|
def get_summand_signture_function(theta):
|
|
|
|
patt_k = knot_as_k_values[-1]
|
|
|
|
# theta should not be larger than k for the pattern.
|
|
theta %= (2 * abs(patt_k) + 1)
|
|
theta = min(theta, 2 * abs(patt_k) + 1 - theta)
|
|
|
|
twisted_part = self.get_blanchfield_for_pattern(patt_k, theta)
|
|
untwisted_part = self.get_untwisted_part(*knot_as_k_values,
|
|
theta=theta)
|
|
return twisted_part, untwisted_part
|
|
get_summand_signture_function.__doc__ = \
|
|
get_summand_signture_function_docsting
|
|
|
|
return get_summand_signture_function
|
|
|
|
def is_metabolizer(self, theta):
|
|
# Check if square alternating difference
|
|
# divided by last q value is integer.
|
|
result = sum(el^2 / self.patt_q_list[idx] * (-1)^idx
|
|
for idx, el in enumerate(theta))
|
|
# for idx, el in enumerate(theta):
|
|
# old_sum += (el^2 / self.patt_q_list[idx] * (-1)^idx)
|
|
return result.is_integer()
|
|
|
|
def is_signature_big_in_ranges(self, ranges_list):
|
|
|
|
for thetas in it.product(*ranges_list):
|
|
|
|
# Check only non-zero metabolizers.
|
|
if not self.is_metabolizer(thetas) or not any(thetas):
|
|
continue
|
|
|
|
signature_is_small = True
|
|
# Check if any element generated by thetas vector
|
|
# has a large signature.
|
|
for shift in range(1, self.q_order):
|
|
shifted_thetas = [shift * th for th in thetas]
|
|
sf = self.signature_as_function_of_theta(*shifted_thetas)
|
|
limit = 5 + np.count_nonzero(shifted_thetas)
|
|
extremum = abs(sf.extremum(limit=limit))
|
|
if shift > 1:
|
|
print(shifted_thetas, end=" ")
|
|
print(extremum)
|
|
if extremum > limit:
|
|
signature_is_small = False
|
|
break
|
|
elif shift == 1:
|
|
print("*" * 10)
|
|
print(shifted_thetas, end=" ")
|
|
print(extremum)
|
|
if signature_is_small:
|
|
print("\n" * 10 + "!" * 1000)
|
|
return False
|
|
return True
|
|
|
|
def is_signature_big_for_all_metabolizers(self):
|
|
num_of_summands = len(self.knot_sum)
|
|
if num_of_summands % 4:
|
|
f_name = self.is_signature_big_for_all_metabolizers.__name__
|
|
msg = "Function {}".format(f_name) + " is implemented only for " +\
|
|
"knots that are direct sums of 4n direct summands."
|
|
raise ValueError(msg)
|
|
|
|
for shift in range(0, num_of_summands, 4):
|
|
ranges_list = num_of_summands * [range(0, 1)]
|
|
ranges_list[shift : shift + 3] = \
|
|
[range(0, i + 1) for i in self.patt_k_list[shift: shift + 3]]
|
|
ranges_list[shift + 3] = range(0, 2)
|
|
if not self.is_signature_big_in_ranges(ranges_list):
|
|
return False
|
|
else:
|
|
print("\nOK")
|
|
return True
|
|
|
|
|
|
|
|
def mod_one(n):
|
|
return n - floor(n)
|
|
|
|
|
|
# CableSum.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)'
|
|
# """
|
|
|
|
CableSum.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.jumps_counter.
|
|
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_summand_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_summand_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)
|
|
+ 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
|
|
"""
|
|
|
|
CableSum.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)
|
|
"""
|
|
|
|
CableSum.get_summand_signature_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
|
|
"""
|