Annals of Combinatorics

, 15:127 | Cite as

Unsigned State Models for the Jones Polynomial

Article

Abstract

It is a well-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 S 3 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 to the Jones and Bollobás-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.

Mathematics Subject Classification

05C10 57M27 

Keywords

Bollobás-Riordan polynomial embedded graphs Jones polynomial medial graph Potts model ribbon graphs vertex model 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWest WaterlooCanada
  2. 2.Department of Mathematics and StatisticsUniversity of South AlabamaMobileUSA

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