A complete knot invariant from contact homology

TL;DR

This paper develops an enhanced contact homology invariant that fully determines knots up to smooth isotopy by capturing the knot group and peripheral subgroup, providing a new proof via holomorphic curves.

Contribution

It introduces a noncommutative Legendrian contact homology that encodes complete knot information, advancing knot invariants from contact geometry.

Findings

Reconstructs the knot group and peripheral subgroup from contact homology.

Provides a holomorphic-curve proof that the conormal torus classifies knots.

Establishes a new, complete knot invariant from contact homology.

Abstract

We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of the conormal torus is a complete knot invariant.