Sutured contact homology, conormal stops and hyperbolic knots

TL;DR

This paper demonstrates that the sutured Legendrian contact homology of a unit fiber in the conormal bundle of a hyperbolic knot uniquely identifies the knot up to mirror, providing a new invariant in contact topology.

Contribution

It introduces a novel application of conormal constructions to hyperbolic knots, establishing sutured Legendrian contact homology as a complete knot invariant.

Findings

Sutured Legendrian contact homology distinguishes hyperbolic knots up to mirror.

The homology of the fiber in the conormal bundle is computed and shown to be a complete invariant.

An explicit relationship between conormal complements and unit bundles is established.

Abstract

We apply the conormal construction to a hyperbolic knot $K \subset S^3$, and study the sutured contact manifold $(V, \xi)$ obtained by taking the complement of a standard neighbourhood of the unit conormal $\La_K \subset (ST^*S^3, \xi_\text{st})$. We show that the sutured Legendrian contact homology of a unit fiber $\La_0$, with its product structure, is a complete invariant of the knot (up to mirror). This can also be seen as the computation of the homology of the fiber in $ST^* S^3$, stopped at $\La_K$. Our main tool is, for any submanifold $N \subset M$, an explicit relationship between the complement of a unit conormal $\La_N$, and the unit bundle of $M \setminus N$.