Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials

A. Anokhina; D. Korzun; E. Lanina; A. Morozov

doi:10.1134/s0040577926010022

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arXiv · 2507.03116

Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials

A. Anokhina·D. Korzun·E. Lanina·A. Morozov

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Abstract

The Goeritz matrix is an alternative to the Kauffman bracket and, in addition, makes it possible to calculate the Jones polynomial faster with some minimal choice of a checkerboard surface of a link diagram. We introduce a modification of the Goeritz method that generalizes the Goeritz matrix for computing the HOMFLY-PT polynomials for any $N$ in the special case of bipartite links. Our method reduces to purely algebraic operations on matrices, and therefore, it can be easily implemented as a computer program. Bipartite links form a rather large family including a special class of Montesinos links constructed from the so-called rational tangles. We demonstrate how to obtain a bipartite diagram of such links and calculate the corresponding HOMFLY-PT polynomials using our developed generalized Goeritz method.

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A. Anokhina, D. Korzun, E. Lanina, A. Morozov. 2025-07-03. Analogue of Goeritz matrices for computation of bipartite HOMFLY-PT polynomials. https://doi.org/10.1134/s0040577926010022

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