Khovanov homology from Floer cohomology

TL;DR

This paper establishes a deep connection between Khovanov homology and symplectic Floer cohomology by proving formality and isomorphism results, confirming a conjecture relating these two invariants of links.

Contribution

It proves the formality of symplectic cup and cap bimodules and constructs a long exact triangle, linking symplectic and combinatorial arc algebras and confirming their isomorphism over integers.

Findings

Khovanov homology equals symplectic Khovanov cohomology in characteristic zero

Symplectic cup and cap bimodules are formal over the field k

Symplectic and combinatorial arc algebras are isomorphic over integers

Abstract

This paper realises the Khovanov homology of a link in the 3-sphere as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k of characteristic zero. Here we prove the symplectic cup and cap bimodules which relate different symplectic arc algebras are themselves formal over k, and construct a long exact triangle for symplectic Khovanov cohomology. We then prove the symplectic and combinatorial arc algebras are isomorphic over the integers in a manner compatible with the cup bimodules. It follows that Khovanov homology and symplectic Khovanov cohomology co-incide in characteristic zero.