Invariants of Legendrian and transverse knots in the standard contact space

Abstract

1.1. The standard contact structure in 3-space, which arises from the identification of R3 with the manifold of l-jets of smooth real functions of one variable, naturally distinguishes two major classes of smooth immersed spatial curves: Legendrian curves, which are integral curves of the contact distribution, that is everywhere tangent to the distribution, and transverse curves, which are nowhere tangent to it. Closed embedded Legendrian and transverse curves are called Legendrian and transverse knots. Theories of Legendrian and transverse knots, which are clearly related to each other, are parallel to the classical knot theory in space. Legendrian and transverse knots have become very popular in contact geometry since the seminal work of Bennequin [3], published in 1983. For Legendrian knots one introduces two integer-valued Legendrian isotopy invariants. The first measures the rotation of an (oriented) knot with respect to the contact distribution; we call it the Maslov number. The second one, which we call the Bennequin number, is defined as the contact self-linking number of the knot. (For exact definitions of these and subsequent notions see Section 2.) Transverse knots have no Maslov numbers, but also have Bennequin numbers. The main achievement of Bennequin’s paper consists in two inequalities for these numbers (see Theorem 2.3 below), which imply, in particular, that the Bennequin number of a topologically unknotted Legendrian knot must be always negative. In turn, this gives rise to a construction of an exotic contact structure in R3 (or rather to a proof, that some previously known contact structures in R3 are not diffeomorphic to the standard one). Bennequin and Maslov numbers may be also used for distinguishing Legendrian or transverse isotopy classes of knots within a topological isotopy class. It is very easy to show that any topological knot is isotopic to (actually is Co approximated by) both Legendrian and transverse knots. It is equally easy to construct topologically isotopic Legendrian or transverse knots with different Bennequin and Maslov (in the Legendrian case) numbers. Since no other specifically Legendrian or transverse invariants of knots have been found so far, one may expect that topologically isotopic Legendrian knots with equal Bennequin and Maslov numbers are Legendrian isotopic, and similarly for transverse knots. The results of this article may be regarded as a confirmation of this conjecture.