Knot contact homology

TL;DR

This paper provides a complete computation of knot contact homology for any link in terms of braid presentations, confirming it matches a combinatorial invariant and introducing multiscale flow trees as a key computational tool.

Contribution

It explicitly computes Legendrian homology of conormal lifts for arbitrary links, verifying its equivalence with a combinatorial knot invariant, and introduces multiscale flow trees for the computation.

Findings

Confirmed the equivalence of Legendrian and combinatorial knot contact homology.

Developed the theory of multiscale flow trees for holomorphic disk analysis.

Provided explicit formulas for Legendrian homology in terms of braid presentations.

Abstract

The conormal lift of a link $K$ in $\R^3$ is a Legendrian submanifold $\Lambda_K$ in the unit cotangent bundle $U^* \R^3$ of $\R^3$ with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of $K$, is defined as the Legendrian homology of $\Lambda_K$, the homology of a differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization $\R \times U^*\R^3$ with Lagrangian boundary condition $\R \times \Lambda_K$. We perform an explicit and complete computation of the Legendrian homology of $\Lambda_K$ for arbitrary links $K$ in terms of a braid presentation of $K$, confirming a conjecture that this invariant agrees with a previously-defined combinatorial version of knot contact homology. The computation uses a double degeneration: the braid degenerates toward a multiple cover of the unknot which in turn degenerates to a point. Under the first degeneration, holomorphic disks converge to gradient flow trees with quantum corrections. The combined degenerations give rise to a new generalization of flow trees called multiscale flow trees. The theory of multiscale flow trees is the key tool in our computation and is already proving to be useful for other computations as well.