The Jones polynomial and graphs on surfaces

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.

19 pages, 9 figures, minor changes

References in corpus (2)

• The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial
• Minimal diagrams of classical knots

Cited by in corpus (5)

• The dealternating number and the alternation number of a closed 3-braid
• Introduction to Graph-Link Theory
• Graphical methods establishing nontriviality of state cycle Khovanov homology classes
• Generalized duality for graphs on surfaces and the signed Bollobas-Riordan polynomial
• Dehn filling, volume, and the Jones polynomial