This script calculates signature functions for knots (cable sums).

The script can be run as a sage script from the terminal or used in interactive mode.

A knot (cable sum) is encoded as a list where each element (also a list) corresponds to a cable knot.

To calculate the number of characters for which signature function vanish use the function eval_cable_for_thetas as shown below.

sage: eval_cable_for_thetas(1, 3], [2], [-1, -2], [-3)

T(2, 3; 2, 7) # T(2, 5) # -T(2, 3; 2, 5) # -T(2, 7) Zero cases: 1 All cases: 1225 Zero theta combinations: (0, 0, 0, 0)

sage:

The numbers given to the function eval_cable_for_thetas are k-values for each component/cable in a direct sum.

To calculate signature function for a knot and a theta value, use function get_function_of_theta_for_sum as follow:

sage: signature_function_generator = get_function_of_theta_for_sum([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 sage:

or like below:

sage: print get_function_of_theta_for_sum([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 sage: