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

A. S. Anokhina^abc, D. V. Korzun^de, E. N. Lanina^abcde, A. Yu. Morozov<sup>abcd

a</sup> Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia
^b National Research Centre "Kurchatov Institute", Moscow, Russia
^c Alikhanov Institute for Theoretical and Experimental Physics, Moscow. Russia
^d Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Region, Russia
^e Saint Petersburg State University, St. Petersburg, Russia

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 easily be 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.

Keywords: Goeritz matrix, Jones polynomial, HOMFLY–PT polynomial, Kauffman bracket, bipartite link.

Received: 25.06.2025
Revised: 25.06.2025

DOI: 10.4213/tmf11034

 English version: Theoretical and Mathematical Physics, 2026, 226:1, 21–65

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