Publications mathématiques de l'IHÉS

, Volume 113, Issue 1, pp 97–208

Khovanov homology is an unknot-detector


DOI: 10.1007/s10240-010-0030-y

Cite this article as:
Kronheimer, P.B. & Mrowka, T.S. Publ.math.IHES (2011) 113: 97. doi:10.1007/s10240-010-0030-y


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.

Copyright information

© IHES and Springer-Verlag 2011

Authors and Affiliations

  1. 1.Harvard UniversityCambridgeUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA