Invariants of knots, embeddings and immersions via contact geometry

Abstract

This paper is an overview of the idea of using contact geometry to construct invariants of immersions and embeddings. In particular, it discusses how to associate a contact manifold to any manifold and a Legendrian submanifold to an embedding or immersion. We then discuss recent work that creates invariants of immersions and embeddings using the Legendrian contact homology of the associated Legendrian submanifold. In recent years classical constructions in contact and symplectic geometry has been applied in the study of topological problems. This idea was used by Arnold [2] where he studies the structure of the space of immersed plane curves. More specifically, Arnold uses a lift of an immersed curve in the plane to the unit cotangent bundle of R, which is diffeomorphic to R × S and which carries a natural contact structure. The lift is a Legendrian knot provided the curve is self transverse and Arnold demonstrated that the contact invariants of this Legendrian knot are useful for understanding the classification of plane curves up to regular homotopies restricted in certain ways. Since then others have picked up on Arnold’s perspective to study codimension one immersions in other dimensions [11]. This had been an exciting area of research for some time when Ooguri and Vafa [15] suggested a similar construction had relevance to modern physics (in particular “largeN -dualities”) a new wave of interest in this construction began. In particular, the idea of using holomorphic curves (or “branes”) on a Lagrangian submanifold to study knots in R seemed very promising. Moreover, the development of Symplectic Field Theory by Eliashberg, Givental and Hofer [3] seemed like the ideal theory to count these holomorphic curves. Recently [6] has provided the analytic underpinnings of the small part of this theory, called Legendrian contact homology in 1-jet spaces, that is necessary to define invariants of knots. The construction of Ooguri and Vafa has inspired Ng [12, 13, 14] to define an invariant of knots using a braid description of the knot. This invariant should compute the contact homology of the Legendrian submanifold associated to a knot in R, though it is ongoing research to prove this. Non the less, Ng’s invariant has been shown to be amazingly powerful [14]. For example, it seems to contain the Alexander and A polynomials, can distinguish the unknot from other knots and at the moment it seems possible that it is a complete invariant of knots. The goal of this paper is to give a somewhat topologist friendly introduction to this theory and describe the connections between the invariant described by Ng and Legendrian contact homology. The story involves a beautiful interplay between topology, geometry, analysis, combinatorics and algebra. We will only begin the story and hopefully provide the background necessary for the reader to pick up [14] or [6] and learn more about it. In particular the two papers concentrate on different