The Jones Polynomial from a Goeritz Matrix

TL;DR

This paper presents an algorithm to compute the Jones polynomial from a Goeritz matrix, extending the method to links in thickened surfaces and developing a theory for cographic matroids.

Contribution

It introduces a new algorithm linking Goeritz matrices to the Jones polynomial and extends the theory to cographic matroids and links in surfaces.

Findings

Algorithm for calculating Kauffman bracket from Goeritz matrix

Jones polynomial recovery from Goeritz matrix in orientable cases

Extension of bracket polynomial to symmetric integer matrices

Abstract

We give an explicit algorithm for calculating the Kauffman bracket of a link diagram from a Goeritz matrix for that link. Further, we show how the Jones polynomial can be recovered from a Goeritz matrix when the corresponding checkerboard surface is orientable, or when more information is known about its Gordon-Litherland form. In the process we develop a theory of Goeritz matrices for cographic matroids, which extends the bracket polynomial to any symmetric integer matrix. We place this work in the context of links in thickened surfaces.