Invariants of Knots, Embeddings and Immersions via Contact Geometry

TL;DR

This paper reviews how contact geometry can be used to create invariants for knots, embeddings, and immersions by associating Legendrian submanifolds and using Legendrian contact homology.

Contribution

It provides an overview of methods linking contact geometry to invariants of immersions and embeddings, highlighting recent advances in Legendrian contact homology.

Findings

Construction of contact invariants for immersions and embeddings

Use of Legendrian contact homology to distinguish knots and embeddings

Framework connecting contact geometry with topological invariants

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.