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Abstract <p> 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(N\)</EquationSource> </InlineEquation> 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. </p>

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

  • 期刊论文

  • 发表日期：

    2026年1月26日

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  • A. S. Anokhina,
  • D. V. Korzun,
  • E. N. Lanina,
  • A. Yu. Morozov

摘要

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.