marchwicka/signature_function
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maria 2021-07-22 23:41:55 +02:00
parent 10c79cb47c
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gaknot/main.sage
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@ -40,7 +40,7 @@ cs = import_sage('cable_signature', package=package, path=path)
# self.f_results = os.path.join(os.getcwd(), "results.out") # self.f_results = os.path.join(os.getcwd(), "results.out")
class Schema: class Schema:
r"""This class stores inreresting schema of cable sums. r"""This class stores interesting schema of cable sums.
Cable knots sum can be given as a scheme, e.g. a scheme from the paper: Cable knots sum can be given as a scheme, e.g. a scheme from the paper:
K(p_1 , p_2 , q_1 , q_2 , q_3 ) = K(p_1 , p_2 , q_1 , q_2 , q_3 ) =
@ -60,24 +60,24 @@ class Schema:
See k_vector setter in class CableTemplate in cable_signature.sage module. See k_vector setter in class CableTemplate in cable_signature.sage module.
Remark 1 Remark 1
In the paper we used p_i and q_i to describe torus knots and cables. In the paper, we used p_i and q_i to describe torus knots and cables.
It was convinient for writing, but in all the code and documentation It was convenient for writing, but in all the code and documentation
only 'q' letter is used to encode torus knots or cables. only 'q' letter is used to encode torus knots or cables.
Remark 2 Remark 2
There are two ways to set k[i] values for a scheme: There are two ways to set k[i] values for a scheme:
via q_vector or via k_vector. via q_vector or k_vector.
Both should be lists and the relation is q[i] = 2 * k[i] + 1, Both should be lists and the relation is q[i] = 2 * k[i] + 1,
i.e. q should be an odd prime and k should be an even number such that i.e. q should be an odd prime and k should be an even number such that
2 * k + 1 is prime. 2 * k + 1 is prime.
To fill the scheme listed above we shoud use a list of lenght 8, To fill the scheme listed above we should use a list of length 8,
and k[0] will be ommited as it is not used in the scheme. and k[0] will be omitted as it is not used in the scheme.
Remark 3 Remark 3
Except for development purposes, q_vector was computed with Except for development purposes, q_vector was computed with
a methode CableTemplate.get_q_vector and flag slice=True. a method CableTemplate.get_q_vector and flag slice=True.
The reason for that is that we were interested only in cases The reason for that is that we were interested only in cases
where a specific relation for each cabling-level is preserved. where a specific relation for each cabling level is preserved.
Consider a cable T(2, q_0; 2, q_1; ...; 2, q_n). Consider a cable T(2, q_0; 2, q_1; ...; 2, q_n).
Then for every q_i, q_(i + 1): q_(i + 1) > q_i * 4. Then for every q_i, q_(i + 1): q_(i + 1) > q_i * 4.