Unsigned state models for the Jones polynomial

TL;DR

This paper demonstrates that the Jones polynomial can be represented as a vertex model of unsigned embedded graphs, broadening the understanding of its combinatorial and topological interpretations.

Contribution

It introduces a novel construction of the Jones polynomial as a state model of unsigned graphs, linking it to embedded graph theory and virtual links.

Findings

Jones polynomial expressed as vertex model of unsigned graphs

Existence of alternating link diagrams with same Kauffman bracket

Connections between Jones and Bollobás-Riordan polynomials

Abstract

It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in the 3-sphere there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating the Jones and Bollobas-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.