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

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

TL;DR

This work extends the Goeritz matrix formalism to compute bipartite HOMFLY-PT polynomials by introducing a four-parameter quaternary Goeritz matrix and an accompanying planar-decomposition-based algorithm. The approach yields purely algebraic, matrix-based operations that can be efficiently implemented computationally, connecting the Kauffman bracket with matrix transformations and providing a corrected normalization for the Jones case. It then generalizes to the HOMFLY-PT setting via a function $\mathcal{M}$ acting on quaternary matrices, with retouching to handle bipartite self-intersections and multiple braids between region pairs. The authors demonstrate the method on twist knots, two-strand torus links, pretzel, rational, and Montesinos knots, including families with parametrized bipartite tangles, and derive both explicit and parametric forms for their HOMFLY-PT polynomials, highlighting the practical utility and scalability of the framework.

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.

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

TL;DR

This work extends the Goeritz matrix formalism to compute bipartite HOMFLY-PT polynomials by introducing a four-parameter quaternary Goeritz matrix and an accompanying planar-decomposition-based algorithm. The approach yields purely algebraic, matrix-based operations that can be efficiently implemented computationally, connecting the Kauffman bracket with matrix transformations and providing a corrected normalization for the Jones case. It then generalizes to the HOMFLY-PT setting via a function acting on quaternary matrices, with retouching to handle bipartite self-intersections and multiple braids between region pairs. The authors demonstrate the method on twist knots, two-strand torus links, pretzel, rational, and Montesinos knots, including families with parametrized bipartite tangles, and derive both explicit and parametric forms for their HOMFLY-PT polynomials, highlighting the practical utility and scalability of the framework.

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 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.

Paper Structure

This paper contains 47 sections, 4 theorems, 112 equations, 41 figures, 1 table.

Table of Contents

  1. Introduction
  2. Kauffman bracket and Jones polynomial: $N=2$
  3. Planar decomposition of a bipartite vertex for any $N$
  4. Text plan.
  5. Obtaining bipartite diagrams
  6. Algorithm for two-strand knots
  7. Remark about twist knots
  8. Bipartiteness of pretzel knots
  9. Rational knots
  10. Montesinos knots
  11. Goeritz matrix
  12. Method for obtaining Jones polynomials from the Goeritz matrix
  13. Preliminary example
  14. Step 1.
  15. Steps 2 and 3.
  16. ...and 32 more sections

Key Result

Theorem 2.1

lewark2016new If all the numbers of crossings in the two-strand tangles $t_1, t_2, t_3$ for a pretzel link $P(t_1, t_2, t_3)$ are odd, and the greatest common divisor satisfies $\gcd(t_1, t_2, t_3) > 1$, then the knot is non-bipartite. If at least one of the numbers $t_1, t_2, t_3$ is even, the link

Figures (41)

  • Figure 1: The Kauffman bracket is the planar decomposition of an $\mathfrak{R}$-matrix vertex in the fundamental representation of $U_q(\mathfrak{sl}_2)$. In this case ($N = 2$), the fundamental representation is isomorphic to its conjugate and therefore the tangles in the diagram have no orientation.
  • Figure 2: Reidemeister moves. There is only single realization of Reidemeister third move (III.R) shown in the figure.
  • Figure 3: Agreement on crossing signs. A positive crossing is indicated by a plus sign, a negative crossing is indicated by a minus sign.
  • Figure 4: Vertical antiparallel lock from anokhina2024planar. Also drawn is the inverse lock made from inverse vertices. Here $\phi = A\{q\}$, $\bar{\phi} = -A^{-1}\{q\}$.
  • Figure 5: If in the expansion of the HOMFLY--PT polynomial in variables $\phi$, $\bar{\phi}$, and $D_N$ we apply the substitution $\phi \rightarrow -q$, $\bar{\phi} \rightarrow -q^{-1}$, $D_N \rightarrow D_2$, then the HOMFLY--PT polynomial reduces to the Jones polynomial of a precursor link. This follows from the equivalence between the planar decomposition of a bipartite vertex (see Fig. \ref{['fig:pladeco']}) and the Kauffman bracket for a single crossing (see Fig. \ref{['fig:Kauff']}). We will refer to bipartite crossings as positive if they correspond to a positive precursor crossing. Furthermore, we call lock tangles vertical or horizontal depending on the orientation of their connecting bar.
  • ...and 36 more figures

Theorems & Definitions (4)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4