ORIENTED STATE MODEL OF THE JONES POLYNOMIAL AND ITS CONNECTION TO THE DICHROMATIC POLYNOMIAL

Abstract

It is well known that Kauffman constructed a state model of the Jones polynomial based on unoriented link diagrams. In his approach, in order to obtain Jones polynomial one needs to calculate both the writhe and the Kauffman bracket. Stimulated by a paper of Altintas (An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput.194 (2007) 168–178), in this paper we present a state sum model based on oriented link diagrams. In our approach, we succeed in adding the writhe to the state sum model and need not to compute the writher any more. We further show that, via our state sum model, Jones polynomial of any link (alternating or not) is a special parametrization of the dichromatic polynomial of a weighted graph with two different edge weights.