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@ -44,8 +44,11 @@ class CableSummand():
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self.knot_description = self.get_summand_descrption(knot_as_k_values)
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self.knot_description = self.get_summand_descrption(knot_as_k_values)
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self.signature_as_function_of_theta = \
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self.signature_as_function_of_theta = \
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self.get_summand_signature_as_theta_function()
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self.get_summand_signature_as_theta_function()
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self.sigma_as_function_of_theta = \
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if verbose:
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self.get_sigma_as_function_of_theta()
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s = self.get_verbose_sigma_as_function_of_theta(verbose=True)
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else:
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s = self.get_sigma_as_function_of_theta()
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self.sigma_as_function_of_theta = s
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@staticmethod
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@staticmethod
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def get_summand_descrption(knot_as_k_values):
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def get_summand_descrption(knot_as_k_values):
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@ -192,14 +195,14 @@ class CableSummand():
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file_name = self.get_file_name_for_summand_plot(theta)
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file_name = self.get_file_name_for_summand_plot(theta)
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save_path = os.path.join(save_path, file_name)
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save_path = os.path.join(save_path, file_name)
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sig.SignaturePloter.plot_sum_of_two(pp, sp, title=title,
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sig.SignaturePloter.plot_sum_of_two(pp, sp, title=title,
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save_path=save_path)
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save_path=save_path)
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def plot_summand(self):
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def plot_summand(self):
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range_limit = min(self.knot_as_k_values[-1] + 1, 3)
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range_limit = min(self.knot_as_k_values[-1] + 1, 3)
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for theta in range(range_limit):
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for theta in range(range_limit):
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self.plot_summand_for_theta(theta)
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self.plot_summand_for_theta(theta)
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def get_sigma_as_function_of_theta(self, verbose=None):
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def get_verbose_sigma_as_function_of_theta(self, verbose=None):
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default_verbose = verbose or self.verbose
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default_verbose = verbose or self.verbose
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last_k = self.knot_as_k_values[-1]
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last_k = self.knot_as_k_values[-1]
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@ -211,28 +214,63 @@ class CableSummand():
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details = details or verbose
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details = details or verbose
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# satellite part (Levine-Tristram signatures)
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# satellite part (Levine-Tristram signatures)
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satellite_part = 0
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satellite_part = 0
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if verbose:
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print(3 * "\n" + 10 * "#" + " " + self.knot_description +
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" " + 10 * "#" + "\n")
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for layer_num, k in enumerate(self.knot_as_k_values[::-1]):
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sigma_q = self.get_untwisted_signature_function(k)
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arg = ksi * theta * layer_num
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sp = sigma_q(arg)
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satellite_part += 2 * sp
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if details and arg:
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label = "ksi * theta * layer_num = " + str(arg)
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title = self.knot_description + ", layer " + str(layer_num)
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title += ", theta = " + str(theta)
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sigma_q.plot(special_point=(mod_one(arg), sp),
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special_label=label,
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title=title,)
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if theta:
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pp = (-last_q + 2 * theta - 2 * (theta^2/last_q)) * sign(last_k)
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else:
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pp = 0
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sigma = pp + satellite_part
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if verbose and theta:
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print(self.knot_description + ", theta = " + str(theta))
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print("pp = " + str(pp), end=', ')
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print("satellite_part = " + str(satellite_part) + "\n")
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if verbose:
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print("sigma({}) = {}".format(self.knot_description,
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pp + satellite_part))
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return pp, satellite_part, sigma
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return sigma_as_function_of_theta
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def get_sigma_as_function_of_theta(self):
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last_k = self.knot_as_k_values[-1]
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last_q = 2 * abs(last_k) + 1
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ksi = 1/last_q
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def sigma_as_function_of_theta(theta):
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satellite_part = 0
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for i, k in enumerate(self.knot_as_k_values[:-1][::-1]):
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for i, k in enumerate(self.knot_as_k_values[:-1][::-1]):
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# print("layer")
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layer_num = i + 1
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layer_num = i + 1
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sigma_q = self.get_untwisted_signature_function(k)
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sigma_q = self.get_untwisted_signature_function(k)
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if details:
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sp = 2 * sigma_q(ksi * theta * layer_num)
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pass
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# print(sigma_q(ksi * theta * layer_num))
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# print(sigma_q)
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sp = 2 * sigma_q(ksi * theta * layer_num) * sign(k)
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satellite_part += sp
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satellite_part += sp
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if theta:
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if theta:
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pp = (-last_q + 2 * theta - 2 * (theta^2/last_q)) * sign(last_k)
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pp = (-last_q + 2 * theta - 2 * (theta^2/last_q)) * sign(last_k)
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else:
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else:
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pp = 0
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pp = 0
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if verbose:
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return pp, satellite_part, pp + satellite_part
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print(self.knot_description + ", theta = " + str(theta))
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print("pp = " + str(pp), end=', ')
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print("satellite_part = " + str(satellite_part))
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return pp + satellite_part
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return sigma_as_function_of_theta
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return sigma_as_function_of_theta
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@ -297,8 +335,8 @@ class CableSum():
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else:
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else:
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save_path = None
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save_path = None
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for i, knot in enumerate(self.knot_summands):
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for theta, knot in zip(thetas, self.knot_summands):
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knot.plot_summand_for_theta(thetas[i], save_path=save_path)
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knot.plot_summand_for_theta(thetas, save_path=save_path)
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# pp, sp, sf = self.signature_as_function_of_theta(*thetas)
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# pp, sp, sf = self.signature_as_function_of_theta(*thetas)
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# title = self.knot_description + ", thetas = " + str(thetas)
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# title = self.knot_description + ", thetas = " + str(thetas)
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@ -358,14 +396,36 @@ class CableSum():
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description = description[:-2] + ") # "
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description = description[:-2] + ") # "
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return description[:-3]
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return description[:-3]
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def get_sigma_as_function_of_theta(self):
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def get_sigma_as_function_of_theta(self, verbose=None):
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def sigma_as_function_of_theta(*thetas, **kwargs):
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default_verbose = verbose or self.verbose
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def sigma_as_function_of_theta(*thetas, verbose=None, **kwargs):
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verbose = verbose or default_verbose
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thetas = self.parse_thetas(*thetas)
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thetas = self.parse_thetas(*thetas)
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result = 0
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sigma_list = []
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for i, knot in enumerate(self.knot_summands):
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sigma_of_th = knot.sigma_as_function_of_theta
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for theta, knot in zip(thetas, self.knot_summands):
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result += sigma_of_th(thetas[i])
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if theta:
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return result
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sigma_of_th = knot.sigma_as_function_of_theta
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sigma_list.append(sigma_of_th(theta))
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else:
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sigma_list.append((0,0,0))
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if verbose:
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print(100 * "*")
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print("Calculation summary for a cable sum:\n" +
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self.knot_description)
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for i, knot in enumerate(self.knot_summands):
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if thetas[i]:
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print("{}. {}".format(i, knot.knot_description))
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print("Pattern part = {}".format(sigma_list[i][0]))
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# print("Pattern part = -{}".format(knot_sum.))
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print("Satellite part = {}".format(sigma_list[i][1]))
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print("Sigma = {}\n".format(sigma_list[i][2]))
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return sum(r[2] for r in sigma_list)
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return sigma_as_function_of_theta
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return sigma_as_function_of_theta
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def get_signature_as_function_of_theta(self, **key_args):
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def get_signature_as_function_of_theta(self, **key_args):
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@ -386,11 +446,17 @@ class CableSum():
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pattern_part = sig.SignatureFunction()
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pattern_part = sig.SignatureFunction()
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# for each cable knot (summand) in cable sum apply theta
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# for each cable knot (summand) in cable sum apply theta
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for i, knot in enumerate(self.knot_summands):
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# for i, knot in enumerate(self.knot_summands):
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sfth = knot.signature_as_function_of_theta
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# sfth = knot.signature_as_function_of_theta
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pp, sp, _ = sfth(thetas[i])
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# pp, sp, _ = sfth(thetas[i])
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# pattern_part += pp
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# satellite_part += sp
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for theta, knot in zip(thetas, self.knot_summands):
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pp, sp, _ = knot.signature_as_function_of_theta(theta)
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pattern_part += pp
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pattern_part += pp
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satellite_part += sp
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satellite_part += sp
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sf = pattern_part + satellite_part
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sf = pattern_part + satellite_part
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if verbose:
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if verbose:
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48840
main.ipynb
48840
main.ipynb
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