marchwicka/periodicity_of_links
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meaning of -1 in output file

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Maria Marchwicka 2018-10-10 15:35:12 +02:00
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README.md
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This repository contains a SageMath script used for computations in [Finite orthogonal groups and periodicity of links](https://arxiv.org/abs/1810.03881v1).
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The following criteria for knots periodicity were implemented:
- Przytycki's criterion for HOMFLYPT polynomials,
- Murasugi's criterion,
- Naik's homological criterion,
- criterion introduced in Theorem 1.4. in above mentioned paper (called in the script and the following description *Borodzik's criterion*).
- Naik's homological criterion (in two steps - called in the script and the following description as Naik's 1 and Naik's 2),
- criterion introduced in Theorem 1.4. in above mentioned paper (called *Borodzik's criterion*).
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For a given list of knots [encoded in PD-code](http://katlas.org/wiki/Planar_Diagrams) (an input file is required) the script generates output with results for each criterion.
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### Results
All results will be saved in a file `results.out`. A line, for example `12a100,3,1,1,1,0,1`, should be read as follow. A knot 12a100 in tests for a period equal 3 (second position) gives results saved in five numbers: `1,1,1,0,1`, which correspond to criteria: Murasugi's, Naik's - part 1, Naik's - part 2, Borodzik's, Przytycki's. Only `0` means that results excluded periodicity. As computations of first 4 numbers depend on previous results, ones zero occurs, the following criteria will be skipped. The Przytycki's criterion is calculated independently of the others. `-1` means that criterion doesn't exclude periodicity, but the criterion wasn't applicable or that the testing algorithm met an empty list of arguments to be verified.
All results will be saved in a file `results.out`. A line, for example `12a100,3,1,1,1,0,1`, should be read as follow. A knot 12a100 in tests for a period equal 3 (second position) gives results saved in five numbers: `1,1,1,0,1`, which correspond to criteria: Murasugi's, Naik's 1, Naik's 2, Borodzik's, Przytycki's. `0` means that results excluded periodicity. As computations of the first 4 numbers depend on previous results, ones zero occurs, the following criteria will be skipped. The Przytycki's criterion is calculated independently of the others. `-1` means that a criterion doesn't exclude periodicity, but it wasn't applicable (like Przytycki's criterion) or that the testing algorithm met an empty list of arguments to be verified (Naik's 2 and Borodzik's criteria). Przytycki's criterion in general is not applicable for non-prime periods, but the corresponding result is set to be `-1` also if no HOMFLYPT polynomial could be found (nigher in an input file nor by using `libhomfly` module).