Khovanov homology is an unknot-detector
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- Kronheimer, P.B. & Mrowka, T.S. Publ.math.IHES (2011) 113: 97. doi:10.1007/s10240-010-0030-y
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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.