The Jones polynomial and graphs on surfaces

2006/05/31 by Oliver T. Dasbach, David Futer, Efstratia Kalfagianni +2
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #math.CO #math.GT #msc:57M25 #semigroups and automata theory

paper · pdf · doi:10.1016/j.jctb.2007.08.003

published as J. Comb. Theory, Series B, Vol 98/2, 2008, pp 384-399 · 19 pages, 9 figures, minor changes

arxiv created 2007/07/24 · openalex publication_date 2007/09/26 · arxiv updated 2009/12/01 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/02

Abstract

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph associated to a link projection. We give some applications of this approach.

Citations

• The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial
• The Kauffman bracket and the Bollobas-Riordan polynomial of ribbon graphs
• Modern Graph Theory
• State models and the jones polynomial
• On the head and the tail of the colored Jones polynomial