, Volume 113, Issue 1, pp 97-208
Date: 11 Feb 2011

Khovanov homology is an unknot-detector

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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.

The work of the first author was supported by the National Science Foundation through NSF grants DMS-0405271 and DMS-0904589.
The work of the second author was supported by NSF grant DMS-0805841.