Khovanov homology is an unknot-detector

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 t...

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Bibliographic Details
Published in:	Publications mathématiques. Institut des hautes études scientifiques Vol. 113; no. 1; pp. 97 - 208
Main Authors:	Kronheimer, P. B., Mrowka, T. S.
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer-Verlag 01-06-2011 Springer
Subjects:	Algebra Algebraic topology Analysis Exact sciences and technology General mathematics General, history and biography Geometry Mathematics Mathematics and Statistics Number Theory Sciences and techniques of general use Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds Modulus Space Spectral Sequence KHOVANOV Homology Hopf Link Floer Homology Invariant Floer homology Invariance principle Rank Cohomology Knot Spectral sequence
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Summary:	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.
ISSN:	0073-8301 1618-1913
DOI:	10.1007/s10240-010-0030-y