Khovanov homology from Floer cohomology

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.

arXiv comment77 pages, 16 figures. v2 (which dates from Dec 2017) corrects one topological hypothesis in the construction of the nc vector field (cf. Remark 3.2) and implements numerous other minor clarifications. This version to appear in JAMS