Khovanov homology is an unknot-detector
Name
Mrowka_Khovanov homology (arxiv).pdf
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1.09 MB
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Author(s)
Kronheimer, P. B.
Mrowka, Tomasz S.
Date Issued
November 2011
Journal
Publications Mathématiques de L'IHÉS
Publisher
Springer-Verlag
Citation
Kronheimer, P. B., and T. S. Mrowka. “Khovanov Homology Is an Unknot-detector.” Publications mathématiques de l’IHÉS 113.1 (2011): 97–208. Web. 27 Apr. 2012. © 2011 IHES and Springer-Verlag
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1007/s10240-010-0030-y
