On the degree of the Jones polynomial

Abstract

In this paper we use the orientation of a link to introduce an additional structure on Kauffman's state model of the Jones polynomial. Although this structure is extremely simple it immediately leads to new results about link diagrams. In this way we sharpen the bounds of Jones on the degrees of the Jones polynomial.We also formulate a conjecture about planar graphs, which implies the well known unknotting conjecture of Milnor.