When the theories meet: Khovanov homology as Hochschild homology of links, arXiv:math.GT/0509334

When the theories meet: Khovanov homology as Hochschild homology of links, arXiv:math.GT/0509334

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by Jozef H. Przytycki

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@MISC{Przytycki_whenthe,
    author = {Jozef H. Przytycki},
    title = {When the theories meet: Khovanov homology as Hochschild homology of links, arXiv:math.GT/0509334},
    year = {}
}

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Abstract

ABSTRACT. We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2,n)-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of Khovanov-Rozansky, sl(n), homology and their deformations we conjecture the connection. The best framework to explore our ideas is to use a comultiplicationfree version of Khovanov homology for graphs developed by L. Helme-Guizon and Y. Rong and extended here to M-reduced case, and in the case of a polygon to noncommutative algebras. In this framework we prove that for any unital algebra A the Hochschild homology of A is isomorphic to graph homology over A of a polygon. We expect that this

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