Khovanov homology is an unknot-detector

Abstract

We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot.

124 pages, 13 figures

References in corpus (4)

• Floer homology and knot complements
• On the Khovanov and knot Floer homologies of quasi-alternating links
• Knots, sutures and excision
• Does Khovanov homology detect the unknot?