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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.
@article{PMIHES_2011__113__97_0,
author = {Kronheimer, P. B. and Mrowka, T. S.},
title = {Khovanov homology is an unknot-detector},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {97--208},
publisher = {Springer-Verlag},
volume = {113},
year = {2011},
doi = {10.1007/s10240-010-0030-y},
mrnumber = {2805599},
zbl = {1241.57017},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-010-0030-y/}
}
TY - JOUR AU - Kronheimer, P. B. AU - Mrowka, T. S. TI - Khovanov homology is an unknot-detector JO - Publications Mathématiques de l'IHÉS PY - 2011 SP - 97 EP - 208 VL - 113 PB - Springer-Verlag UR - http://geodesic.mathdoc.fr/articles/10.1007/s10240-010-0030-y/ DO - 10.1007/s10240-010-0030-y LA - en ID - PMIHES_2011__113__97_0 ER -
%0 Journal Article %A Kronheimer, P. B. %A Mrowka, T. S. %T Khovanov homology is an unknot-detector %J Publications Mathématiques de l'IHÉS %D 2011 %P 97-208 %V 113 %I Springer-Verlag %U http://geodesic.mathdoc.fr/articles/10.1007/s10240-010-0030-y/ %R 10.1007/s10240-010-0030-y %G en %F PMIHES_2011__113__97_0
Kronheimer, P. B.; Mrowka, T. S. Khovanov homology is an unknot-detector. Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 97-208. doi: 10.1007/s10240-010-0030-y
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