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This script calculates signature functions for knots (cable sums).
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The script can be run as a sage script from the terminal or used in interactive mode.
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A knot (cable sum) is encoded as a list where each element (also a list) corresponds to a cable knot.
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To calculate the number of characters for which signature function vanish use the function eval_cable_for_thetas as shown below.
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sage: eval_cable_for_thetas([[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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Zero cases: 1
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All cases: 1225
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Zero theta combinations:
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(0, 0, 0, 0)
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sage:
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The numbers given to the function eval_cable_for_thetas are k-values for each
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component/cable in a direct sum.
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To calculate signature function for a knot and a theta value, use function
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get_function_of_theta_for_sum as follow:
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sage: signature_function_generator = get_function_of_theta_for_sum([1, 3], [2], [-1, -2], [-3])
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sage: sf = signature_function_generator(2, 1, 2, 2)
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sage: print sf
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0: 0
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5/42: 1
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1/7: 0
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1/5: -1
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7/30: -1
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2/5: 1
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3/7: 0
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13/30: -1
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19/42: -1
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23/42: 1
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17/30: 1
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4/7: 0
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3/5: -1
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23/30: 1
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4/5: 1
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6/7: 0
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37/42: -1
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sage:
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or like below:
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sage: print get_function_of_theta_for_sum([1, 3], [2], [-1, -2], [-3])(2, 1, 2, 2)
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0: 0
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1/7: 0
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1/6: 0
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1/5: -1
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2/5: 1
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3/7: 0
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1/2: 0
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4/7: 0
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3/5: -1
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4/5: 1
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5/6: 0
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6/7: 0
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sage:
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