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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.

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Authors and Affiliations

  1. Harvard University, Cambridge, MA, 02138, USA

    P. B. Kronheimer

  2. Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    T. S. Mrowka

Authors
  1. P. B. Kronheimer
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  2. T. S. Mrowka
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Corresponding author

Correspondence to P. B. Kronheimer.

Additional information

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.

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Kronheimer, P.B., Mrowka, T.S. Khovanov homology is an unknot-detector. Publ.math.IHES 113, 97–208 (2011). https://doi.org/10.1007/s10240-010-0030-y

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  • DOI: https://doi.org/10.1007/s10240-010-0030-y

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