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    A geometric construction of colored HOMFLYPT homology

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    The aim of this paper is two-fold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov-Rozansky. Our method is to construct this invariant in terms of the cohomology of various sheaves on certain algebraic groups, in the same spirit as the authors' previous work on Soergel bimodules. All the differentials and gradings which appear in the construction of HOMFLYPT homology are given a geometric interpretation. In fact, with only minor modifications, we can extend this construction to give a categorification of the colored HOMFLYPT polynomial, colored HOMFLYPT homology. We show that it is in fact a knot invariant categorifying the colored HOMFLYPT polynomial and that this coincides with the categorification proposed by Mackaay, Stosic and Vaz.

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    Description

    Title : A geometric construction of colored HOMFLYPT homology
    Author(s) : Ben Webster, Geordie Williamson
    Abstract : The aim of this paper is two-fold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov-Rozansky. Our method is to construct this invariant in terms of the cohomology of various sheaves on certain algebraic groups, in the same spirit as the authors' previous work on Soergel bimodules. All the differentials and gradings which appear in the construction of HOMFLYPT homology are given a geometric interpretation. In fact, with only minor modifications, we can extend this construction to give a categorification of the colored HOMFLYPT polynomial, colored HOMFLYPT homology. We show that it is in fact a knot invariant categorifying the colored HOMFLYPT polynomial and that this coincides with the categorification proposed by Mackaay, Stosic and Vaz.
    Subject : unspecified
    Area : Other
    Language : English
    Year : 2009

    Affiliations Mathematics, Northeastern University, Boston
    Journal : arXiv
    Pages : 31
    Url : http://arxiv.org/abs/0905.0486

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