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README.md First version of README.md, input file with HOMFLYPT polynomials added 2018-10-10 10:46:27 +02:00

README.md

A Sage script to verify periodicity

This repository contains a SageMath script used for computations in Finite orthogonal groups and periodicity of links.

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).

For a given list of knots encoded in PD-code (an input file is required) the script generates output with results for each criterion.

We tested all knots up to 15 crossings and for periods 3, 5, 7, 9, 11, 13.

Running the script

Prerequisites

  • SageMath 8.1.

  • For Przytycki's criterion the module libhomfly is needed to calculate HOMFLYPT polynomial. There is also possibility to read saved polynomials from a file (file homflypt.input in the repository contains HOMFLYPTY polynomials for some knots).

Input data

Files with PD-codes for all knots up to 15 crossing are available in the repository (knots_11_15.txt and knots_3_10.txt) and should be saved in the same folder as the script or a path in the class MySettings should be adjusted. Similarly, if other input files will be used, settings have to be changed. A correction in parsing or reading functions may be also necessary (check_11_to_15(), check_up_to_10(), parse_pd_code(), parse_knot_name()).

Running in terminal

The script should be run in terminal: sage periodicity.sage.

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.

Authors

  • Maria Marchwicka
  • Wojciech Politarczyk

License

This project is licensed under the GNU General Public License - see the LICENSE.md file for details.