Knot contact homology and open Gromov-Witten theory

TL;DR

This paper explores knot contact homology, linking it to Gromov-Witten theory, and demonstrates its role as a complete knot invariant with deep connections to symplectic geometry and physics.

Contribution

It reviews the theory of knot contact homology, establishes it as a complete knot invariant, and connects it to Gromov-Witten potentials and higher genus relations.

Findings

Knot contact homology is a complete knot invariant.

Connections established between augmentation variety and Gromov-Witten disk potentials.

Discussion of higher genus relations quantizing the augmentation variety.

Abstract

Knot contact homology studies symplectic and contact geometric properties of conormals of knots in 3-manifolds using holomorphic curve techniques. It has connections to both mathematical and physical theories. On the mathematical side, we review the theory, show that it gives a complete knot invariant, and discuss its connections to Fukaya categories, string topology, and micro-local sheaves. On the physical side, we describe the connection between the augmentation variety of knot contact homology and Gromov-Witten disk potentials, and discuss the corresponding higher genus relation that quantizes the augmentation variety.