Unsigned state models for the Jones polynomial

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.

Minor corrections. To appear in Annals of Combinatorics

References in corpus (5)

• The Jones polynomial and graphs on surfaces
• The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial
• Knot invariants and the Bollobas-Riordan polynomial of embedded graphs
• Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs
• Chromatic Polynomial, Colered Jones Function and q-Binomial Counting

Cited by in corpus (7)

• Partial duality and Bollobas and Riordan's ribbon graph polynomial
• Separability and the genus of a partial dual
• Bipartite partial duals and circuits in medial graphs
• Partial duals of plane graphs, separability and the graphs of knots
• On the Seifert graphs of a link diagram and its parallels
• Graph polynomials and their applications II: Interrelations and interpretations
• Excluded minors and the ribbon graphs of knots