Khovanov homology is an unknot-detector

  title={Khovanov homology is an unknot-detector},
  author={Peter B. Kronheimer and Tomasz S. Mrowka},
  journal={Publications math{\'e}matiques de l'IH{\'E}S},
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. 
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