A Connection between the Kauffman Polynomial and Thurston-Bennequin Invariant

Abstract

The negative of the highest degree of the framing variable a in the Kauffman polynomial provides a bound for the Thurston-Bennequin number of Legendrian knots in a smooth knot type. This is a less than obvious connection between an invariant defined on smooth knots and an invariant defined only on Legendrian knots. In preparation for this theorem, this paper presents some elementary contact geometry necessary to discuss Legendrian knots and the Thurston-Bennequin number, and provides the somewhat extensive proof that the Kauffman polynomial is a well-defined invariant.