A 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.

v2: 59 pages; to appear in Inventiones. The arXiv version of this paper includes a speculative Section 6 about partially wrapped Floer cohomology, which is omitted from the published version

References in corpus (3)

• Duality between Lagrangian and Legendrian invariants
• Wrapped microlocal sheaves on pairs of pants
• The conormal torus is a complete knot invariant

Cited by in corpus (22)

• Covariantly functorial wrapped Floer theory on Liouville sectors
• Microlocal Morse theory of wrapped Fukaya categories
• Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications
• The conormal torus is a complete knot invariant
• Calabi-Yau structures on topological Fukaya categories
• Relative Calabi-Yau completions
• Sheaves via augmentations of Legendrian surfaces
• Fiber Floer cohomology and conormal stops
• Chekanov-Eliashberg dg-algebras for singular Legendrians
• Geometry and analysis of contact instantons and entanglement of Legendrian links I
• To compute orientations of Morse flow trees in Legendrian contact homology
• Simple Sheaves for Knot Conormals
• A Chekanov-Eliashberg algebra for Legendrian graphs
• The Reidemeister graph is a complete knot invariant
• Formality of Floer complex of the ideal boundary of hyperbolic knot complement
• Nearby Lagrangian fibers and Whitney sphere links
• Tangle contact homology
• Augmentations and link group representations
• Augmentations and sheaves for links
• Legendrian non-isotopic unit conormal bundles in high dimensions
• 2-knot homology and Yoshikawa move
• Perverse Sheaves and Knot Contact Homology