sigma calc

cable_signature.sage

@@ -32,12 +32,15 @@ PLOTS_DIR = "plots"
 
 class CableSummand():
 
+
     def __init__(self, knot_as_k_values):
 
         self.knot_as_k_values = knot_as_k_values
         self.knot_description = self.get_summand_descrption(knot_as_k_values)
         self.signature_as_function_of_theta = \
                                 self.get_summand_signature_as_theta_function()
+        self.sigma_as_function_of_theta = \
+                                self.get_sigma_function()
 
     @staticmethod
     def get_summand_descrption(knot_as_k_values):
@@ -191,6 +194,39 @@ class CableSummand():
         for theta in range(range_limit):
             self.plot_summand_for_theta(theta)
 
+    def get_sigma_function(self):
+
+        last_k = self.knot_as_k_values[-1]
+        last_q = 2 * abs(last_k) + 1
+        ksi = 1/last_q
+
+        def sigma_function(theta, print_results=False):
+            # satellite part (Levine-Tristram signatures)
+            sp = 0
+            for i, k in enumerate(self.knot_as_k_values[:-1][::-1]):
+                # print("layer")
+                layer_num = i + 1
+                sigma_q = self.get_untwisted_signature_function(k)
+                # print(sigma_q(ksi * theta * layer_num))
+                # print(sigma_q)
+                satelit_part = 2 * sigma_q(ksi * theta * layer_num) * sign(k)
+                sp += satelit_part
+            if theta:
+                pp = (-last_q + 2 * theta - 2 * (theta^2/last_q)) * sign(last_k)
+            else:
+                pp = 0
+            result = pp + sp
+            # print("pp = ", pp )
+            # print("sp = ", sp)
+            # print(result)
+            return result
+        return sigma_function
+
+
+
+
+
+
 class CableSum():
 
     def __init__(self, knot_sum):
@@ -210,6 +246,10 @@ class CableSum():
         self.knot_summands = [CableSummand(k) for k in knot_sum]
         self.signature_as_function_of_theta = \
                         self.get_signature_as_function_of_theta()
+        self.sigma_as_function_of_theta = \
+                        self.get_sigma_as_function_of_theta()
+
+
 
     def __call__(self, *thetas):
         return self.signature_as_function_of_theta(*thetas)
@@ -306,6 +346,8 @@ class CableSum():
             description = description[:-2] + ") # "
         return description[:-3]
 
+
+
     def get_signature_as_function_of_theta(self, **key_args):
         if 'verbose' in key_args:
             verbose_default = key_args['verbose']
@@ -403,6 +445,76 @@ class CableSum():
         return True
 
 
+
+    def get_sigma_as_function_of_theta(self):
+        def sigma_as_function_of_theta(*thetas, **kwargs):
+            thetas = self.parse_thetas(*thetas)
+            result = 0
+            for i, knot in enumerate(self.knot_summands):
+                sigma_of_th = knot.sigma_as_function_of_theta
+                result += sigma_of_th(thetas[i])
+            return result
+        return sigma_as_function_of_theta
+
+
+    def is_sigma_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]
+                # pp, sp, sf= self.signature_as_function_of_theta(*shifted_thetas)
+                limit = 5 + np.count_nonzero(shifted_thetas)
+                ext = self.sigma_as_function_of_theta(shifted_thetas)
+                print(ext)
+                extremum = abs(ext)
+                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(ext)
+            if signature_is_small:
+                print("\n" * 10 + "!" * 1000)
+                return False
+        return True
+
+    def is_sigma_big_for_all_metabolizers(self):
+        num_of_summands = len(self.knot_sum_as_k_valus)
+        if num_of_summands % 4:
+            f_name = self.is_sima_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_sigma_big_in_ranges(ranges_list):
+                return False
+            else:
+                print("\nOK")
+        return True
+
+
+
+
+
+
+
 class CableTemplate():
 
     def __init__(self, knot_formula, q_vector=None, k_vector=None,