Estimates for the Bennequin number of Legendrian links from state models for knot polynomials

Abstract

A contact structure in a 3-dimensional manifold is a completely nonintegrable 2-dimensional distribution: if the distribution is the kernel of a (locally defined) 1-form λ then λ∧dλ � 0everywhere. The standard contact structure in 3-space, arising from the identification of R 3 with the space of 1-jets of functions on the line, is given by the contact form λ = dz − yd x; here x is a point on the line, z is the value of a function and y the value of the derivative at x. The same formula defines the standard contact structure in the space of 1-jets of functions on the circle (x being the cyclic coordinate), the space which is topologically the solid torus. In this paper we will be concerned only with these two contact manifolds. In a contact 3-dimensional manifold one considers two classes of knots and links: the transverse and the Legendrian ones. The former are everywhere transverse to the contact distribution, and the latter are everywhere tangent to it. Every topological isotopy class of knots in a contact 3-fold contains a transverse and a Legendrian knot. The main problem of contact knot theory (which has made, so far, only a few first steps) is to classify, up to contact isotopies, transverse and Legendrian knots within each topological isotopy class. In particular, one would like to have specifically contact invariants of transverse and Legendrian knots and links. One such invariant is easily defined. Legendrian and transverse knots in the standard contact space have natural framings given by the vector fields ∂z and ∂y, respectively. The corresponding self-linking number is called the Bennequin number; it is denoted by β(K) where K is a Legendrian or a transverse knot 1 . Similarly one defines the Bennequin number of an oriented Legendrian or transverse link. The study of knots in contact 3-dimensional manifolds was put forward by the seminal paper by D. Bennequin [Be]. One of the main results of this paper