Legendrian Contact Homology in P X R

TL;DR

This paper establishes a rigorous foundation for Legendrian contact homology in contact manifolds of the form P×R, enabling new invariants for submanifolds and immersions in Euclidean spaces, with applications to knot theory.

Contribution

It provides a rigorous framework for Legendrian contact homology in P×R contact manifolds, including applications to isotopy invariants and knot invariants in Euclidean spaces.

Findings

Established contact homology as an isotopy invariant for submanifolds in R^n.

Extended contact homology to invariants of self transverse immersions.

Connected contact homology to new knot invariants for n=3.

Abstract

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form $P\times \R$ where $P$ is an exact symplectic manifold is established. The class of such contact manifolds include 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of $\R^n$ and, more generally, invariants of self transverse immersions into $\R^n$ up to restricted regular homotopies. When $n=3$, this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng