fix th mod q for sigma
This commit is contained in:
parent
68cf72305e
commit
784d677a8d
@ -15,7 +15,6 @@ SIGNATURE = 1
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def import_sage(module_name):
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importlib.invalidate_caches()
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sage_name = module_name + ".sage"
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python_name = module_name + ".sage.py"
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@ -34,7 +33,7 @@ sig = import_sage('signature')
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# 9.15 (9.9)
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PLOTS_DIR = "plots"
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class CableSummand():
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class CableSummand:
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def __init__(self, knot_as_k_values, verbose=False):
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@ -44,11 +43,7 @@ class CableSummand():
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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.get_summand_signature_as_theta_function()
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if verbose:
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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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self.sigma_as_function_of_theta = self.get_sigma_as_function_of_theta()
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@staticmethod
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def get_summand_descrption(knot_as_k_values):
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@ -138,28 +133,38 @@ class CableSummand():
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return satellite_part
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@staticmethod
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def get_untwisted_signature_function(j):
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def get_untwisted_signature_function(k=None, q=None):
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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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if q is not None:
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signum = sign(q)
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q = abs(q)
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k = (q - 1)/2
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elif k is not None:
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signum = sign(k)
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k = abs(k)
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q = 2 * k + 1
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else:
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raise ValueError('k or q value must be given')
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counter = Counter({(2 * a + 1)/(2 * q) : -signum
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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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counter.update(Counter({(2 * a + 1)/(2 * q) : signum
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for a in range(k + 1, q)}))
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return sig.SignatureFunction(counter=counter)
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def get_summand_signature_as_theta_function(self):
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knot_as_k_values = self.knot_as_k_values
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# knot_as_k_values = self.knot_as_k_values
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def get_summand_signture_function(theta):
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patt_k = knot_as_k_values[-1]
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patt_k = self.knot_as_k_values[-1]
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# theta should not be larger than k for the pattern.
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theta %= (2 * abs(patt_k) + 1)
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theta = min(theta, 2 * abs(patt_k) + 1 - theta)
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pattern_part = self.get_blanchfield_for_pattern(patt_k, theta)
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satellite_part = self.get_satellite_part(*knot_as_k_values,
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satellite_part = self.get_satellite_part(*self.knot_as_k_values,
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theta=theta)
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sf = satellite_part + pattern_part
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@ -197,68 +202,85 @@ class CableSummand():
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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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def plot_summand(self):
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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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self.plot_summand_for_theta(theta)
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def plot_summand_sigma(self):
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sigma = self.sigma_as_function_of_theta
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# pattern part
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th_values = list(range(abs(self.knot_as_k_values[-1]) + 1))
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y = [sigma(th)[0] for th in th_values]
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print("plot_summand_sigma")
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print(th_values)
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print(y)
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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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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, verbose=None, details=None):
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verbose = verbose or default_verbose
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details = details or verbose
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# satellite part (Levine-Tristram signatures)
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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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# satellite_part
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patt_k = self.knot_as_k_values[-1]
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patt_q = 2 * abs(patt_k) + 1
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ksi = 1/patt_q
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x = []
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s = self.get_untwisted_signature_function
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list_of_signatue_functions = [s(k) for k in self.knot_as_k_values[:-1]]
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for i, k in enumerate(self.knot_as_k_values[:-1][::-1]):
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layer_num = i + 1
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x.append(ksi * layer_num)
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print("\nx")
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print(x)
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print(th_values)
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print("\nx product")
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x = list(set(it.product(x, th_values)))
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x = [(a * b) for (a, b) in x]
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print(x)
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def print_sigma_as_function_of_theta(self, theta):
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if not theta:
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return
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# theta should not be larger than q for the pattern.
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patt_k = self.knot_as_k_values[-1]
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patt_q = 2 * abs(patt_k) + 1
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theta %= patt_q
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ksi = 1/patt_q
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# satellite part (Levine-Tristram signatures)
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print(3 * "\n" + 10 * "#" + " " + self.knot_description +
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" " + 10 * "#" + "\n")
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satellite_part = 0
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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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pp = (-patt_q + 2 * theta - 2 * (theta^2/patt_q)) * sign(patt_k)
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sigma = 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) + "\n")
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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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patt_k = self.knot_as_k_values[-1]
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patt_q = 2 * abs(patt_k) + 1
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ksi = 1/patt_q
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def sigma_as_function_of_theta(theta):
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if theta == 0:
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return 0, 0, 0
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# theta should not be larger than q for the pattern.
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patt_k = self.knot_as_k_values[-1]
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theta %= (2 * abs(patt_k) + 1)
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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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@ -267,18 +289,14 @@ class CableSummand():
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sp = 2 * sigma_q(ksi * theta * layer_num)
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satellite_part += sp
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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 = (-patt_q + 2 * theta - 2 * (theta^2/patt_q)) * sign(patt_k)
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else:
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pp = 0
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return pp, satellite_part, pp + satellite_part
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return sigma_as_function_of_theta
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class CableSum():
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class CableSum:
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def __init__(self, knot_sum, verbose=False):
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@ -361,9 +379,9 @@ class CableSum():
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return dir_name
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def plot_all_summands(self):
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def plot_sigma_for_summands(self):
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for knot in self.knot_summands:
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knot.plot_summand()
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knot.plot_summand_sigma()
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def parse_thetas(self, *thetas):
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summands_num = len(self.knot_sum_as_k_valus)
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@ -386,6 +404,16 @@ class CableSum():
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@staticmethod
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def get_knot_descrption(knot_sum):
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"""
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Arguments:
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arbitrary number of lists of numbers,
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each list encodes a single cable.
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Examples:
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sage: get_knot_descrption([1, 3], [2], [-1, -2], [-3])
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'T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7)'
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"""
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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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@ -398,34 +426,17 @@ class CableSum():
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def get_sigma_as_function_of_theta(self, verbose=None):
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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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sigma_list = []
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sigma = 0
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for th, knot in zip(thetas, self.knot_summands):
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_, _, s = knot.sigma_as_function_of_theta(th)
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sigma += s
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return sigma
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for theta, knot in zip(thetas, self.knot_summands):
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if theta:
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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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def get_signature_as_function_of_theta(self, **key_args):
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@ -446,13 +457,8 @@ class CableSum():
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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 i, knot in enumerate(self.knot_summands):
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# sfth = knot.signature_as_function_of_theta
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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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for th, knot in zip(thetas, self.knot_summands):
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pp, sp, _ = knot.signature_as_function_of_theta(th)
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pattern_part += pp
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satellite_part += sp
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@ -479,8 +485,6 @@ class CableSum():
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# divided by last q value is integer.
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result = sum(el^2 / self.patt_q_list[idx] * (-1)^idx
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for idx, el in enumerate(theta))
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# for idx, el in enumerate(theta):
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# old_sum += (el^2 / self.patt_q_list[idx] * (-1)^idx)
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return result.is_integer()
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def is_function_big_in_ranges(self, ranges_list, invariant=SIGMA,
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@ -488,42 +492,83 @@ class CableSum():
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verbose = verbose or self.verbose
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if invariant == SIGNATURE:
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get_invariant = self.get_sign_ext_for_theta
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name = "signature (extremum)"
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else:
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get_invariant = self.sigma_as_function_of_theta
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name = "sigma value"
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for thetas in it.product(*ranges_list):
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# Check only non-zero metabolizers.
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if not self.is_metabolizer(thetas) or not any(thetas):
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continue
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#
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# cond1 = thetas[0] and thetas[3] and not thetas[1] and not thetas[2]
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# cond = thetas[0] and thetas[3] and not thetas[1] and not thetas[2]
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function_is_small = True
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# Check if any element generated by thetas vector
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# has a large signature.
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# has a large signature or sigma.
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for shift in range(1, self.q_order):
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shifted_thetas = [shift * th for th in thetas]
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limit = 5 + np.count_nonzero(shifted_thetas)
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# pp, sp, sf= self.signature_as_function_of_theta(*shifted_thetas)
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inv_value = get_invariant(shifted_thetas, limit=limit)
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# print(ext)
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abs_value = abs(inv_value)
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if shift > 1:
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print(shifted_thetas, end=" ")
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print(inv_value)
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elif shift == 1 and verbose:
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print("*" * 10)
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if verbose:
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if shift == 1:
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print("\n" + "*" * 10)
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print("Knot sum:\n" + self.knot_description)
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print("[ characters ] " + name)
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print(shifted_thetas, end=" ")
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print(inv_value)
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if abs_value > limit:
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function_is_small = False
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if invariant == SIGMA and verbose:
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self.print_calculations_for_sigma(*shifted_thetas)
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break
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if function_is_small:
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print("\n" * 10 + "!" * 1000)
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return False
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return True
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def print_calculations_for_sigma(self, *thetas):
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print("Calculation details for a cable sum:\n" +
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self.knot_description + "\nand theta vector: " +
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str(thetas) + "\n")
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for i, (th, knot) in enumerate(zip(thetas, self.knot_summands)):
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print("{}. {}, theta = {}".format(i + 1, knot.knot_description, th))
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if not th:
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continue
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patt_k = knot.knot_as_k_values[-1]
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q = 2 * abs(patt_k) + 1
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th %= q
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if patt_k > 0:
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print("Pattern part = pp")
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else:
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print("Pattern part = -pp")
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print("pp = -q + 2 * theta * (q - theta)/q =")
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print(" = -{} + 2 * {} * ({} - {} )/{} =".format(
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q, th, q, th, q))
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print(" = -{} + {} * ({} )/{} =".format(
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q, 2 * th, q - th, q))
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print(" = -{} + {} * {} = ".format(
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q, 2 * th, (q - th)/ q))
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print(" = -{} + {} = ".format(
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q, 2 * th * (q - th)/ q))
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print(" = {} ".format(
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-q + (2 * th * (q - th)/ q)))
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pp = (-q + 2 * th - 2 * (th^2/q)) * sign(patt_k)
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sigma = knot.sigma_as_function_of_theta(th)
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print("Pattern part = {} ~ {}".format(sigma[0],int(sigma[0])))
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print("Satellite part = {}".format(sigma[1]))
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print("Sigma = {} ~ {}\n".format(sigma[2], int(sigma[2])))
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def is_function_big_for_all_metabolizers(self, invariant=SIGMA):
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num_of_summands = len(self.knot_sum_as_k_valus)
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if num_of_summands % 4:
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@ -539,13 +584,10 @@ class CableSum():
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ranges_list[shift + 3] = range(0, 2)
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if not self.is_function_big_in_ranges(ranges_list, invariant):
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return False
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print("\nOK")
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return True
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class CableTemplate():
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class CableTemplate:
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def __init__(self, knot_formula, q_vector=None, k_vector=None,
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generate_q_vector=True, slice=True, verbose=False):
|
||||
@ -568,8 +610,8 @@ class CableTemplate():
|
||||
self.fill_q_vector()
|
||||
return self._cable
|
||||
|
||||
def fill_q_vector(self, q_vector=None, slice=True):
|
||||
self.q_vector = q_vector or self.get_q_vector(slice)
|
||||
def fill_q_vector(self, q_vector=None, slice=True, lowest_number=2):
|
||||
self.q_vector = q_vector or self.get_q_vector(slice, lowest_number)
|
||||
|
||||
@property
|
||||
def knot_formula(self):
|
||||
@ -604,9 +646,8 @@ class CableTemplate():
|
||||
numbers = map(int, numbers)
|
||||
return max(numbers)
|
||||
|
||||
def get_q_vector(self, slice=True):
|
||||
def get_q_vector(self, slice=True, lowest_number=2):
|
||||
knot_formula = self.knot_formula
|
||||
lowest_number = 2
|
||||
q_vector = [0] * (self.extract_max(knot_formula) + 1)
|
||||
P = Primes()
|
||||
for layer in self.get_layers_from_formula(knot_formula)[::-1]:
|
||||
@ -643,15 +684,6 @@ 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
|
||||
|
||||
Loading…
Reference in New Issue
Block a user