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

TL;DR

The paper introduces a matrix-based method extending the Goeritz matrix to efficiently compute bipartite HOMFLY-PT polynomials, applicable to a broad class of links including Montesinos links.

Contribution

It generalizes the Goeritz matrix approach for bipartite links, enabling algebraic computation of HOMFLY-PT polynomials for any N.

Findings

Method simplifies calculations to algebraic matrix operations.

Successfully applied to bipartite and Montesinos links.

Facilitates computer implementation for polynomial computation.

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.