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The Superpolynomial for Knot Homologies

Dunfield, Nathan M. and Gukov, Sergei and Rasmussen, Jacob (2006) The Superpolynomial for Knot Homologies. Experimental Mathematics, 15 (2). pp. 129-159. ISSN 1058-6458. https://resolver.caltech.edu/CaltechAUTHORS:20160511-091653426

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Abstract

We propose a framework for unifying the sl(N) Khovanov– Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory that categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a family of differentials. Roughly speaking, the triply graded theory by itself captures the large-N behavior of the sl(N) homology, and differentials capture nonstable behavior for small N, including knot Floer homology. The differentials themselves should come from another variant of sl(N) homology, namely the deformations of it studied by Gornik, building on work of Lee. While we do not give a mathematical definition of the triply graded theory, the rich formal structure we propose is powerful enough to make many nontrivial predictions about the existing knot homologies that can then be checked directly. We include many examples in which we can exhibit a likely candidate for the triply graded theory, and these demonstrate the internal consistency of our axioms. We conclude with a detailed study of torus knots, developing a picture that gives new predictions even for the original sl(2) Khovanov homology.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1080/10586458.2006.10128956DOIArticle
http://www.tandfonline.com/doi/abs/10.1080/10586458.2006.10128956PublisherArticle
https://arxiv.org/abs/math/0505662arXivDiscussion Paper
ORCID:
AuthorORCID
Gukov, Sergei0000-0002-9486-1762
Additional Information:© 2006 Taylor & Francis. We are grateful to P. Etingof, B. Gornik, V. Kac, M. Khovanov, C. Manolescu, P. Ozsváth, A. Schwarz, C. Taubes, C. Vafa, and Z. Szabó for valuable discussions. N.D. was partially supported by NSF grant #DMS-0405491 and a Sloan Fellowship. This work was conducted during the period S.G. served as a Clay Mathematics Institute Long-Term Prize Fellow. J.R. was partially supported by an NSF Postdoctoral Fellowship.
Funders:
Funding AgencyGrant Number
NSFDMS-0405491
Alfred P. Sloan FoundationUNSPECIFIED
Clay Mathematics InstituteUNSPECIFIED
NSF Postdoctoral FellowshipUNSPECIFIED
Issue or Number:2
Record Number:CaltechAUTHORS:20160511-091653426
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160511-091653426
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:66977
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:11 May 2016 16:47
Last Modified:09 Mar 2020 13:18

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