The Jones Polynomial from a Goeritz Matrix
Joe Boninger
TL;DR
The paper develops a complete framework connecting Goeritz matrices to the Kauffman bracket and the Jones polynomial, extending the construction to cographic matroids and to links in thickened surfaces. It defines the μ-polynomial for symmetric integer matrices and proves it recovers the Kauffman bracket on Goeritz matrices, while also establishing a precise τ–μ relationship that underpins bracket computations from matroid data. It then shows how the Jones polynomial can be recovered from μ[G] together with Gordon–Litherland data when the checkerboard surface is orientable, and provides extensions to nonorientable cases with additional information. Finally, it introduces ν-invariants for links in thickened surfaces, relates them to determinants, and demonstrates how Goeritz matrices encode rich topological information in these broader settings, including virtual link perspectives. Overall, the work broadens the algebraic-topological toolbox for computing and understanding the Jones polynomial via Goeritz matrices across classical, surface-embedded, and virtual contexts.
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.