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Lev Rozansky : A categorification of the stable Witten-Reshetikhin-Turaev invariant of links in S2 x S1

This work was done in close collaboration with M. Khovanov. The Witten-Reshetikhin-Turaev invariant Z(M,L;r) of a link L in a 3-manifold M is a seemingly random function of an integer r. However, for a small class of 3-manifolds constructed by identical gluing of two handlebodies (e.g., for S3 and for S2 x S1) and for sufficiently large values of r the ratio Z(M,L;r)/Z(M;r) is equal to a rational function J(M,L;q) of q evaluated at the first 2r-th root of unity. If M = S3, then J is the Jones polynomial. Khovanov categorified J(S3,L), that is, to a link L in S3 he assigned a homology H(L) with an extra Z-grading such that its graded Euler characteristic equals J(S3,L). We extend Khovanov's construction to links in S2 x S1 thus categorifying J(S2xS1,L). In his work on categorification of the Jones polynomial, Khovanov introduced special algebras H_n and assigned a H_m x H_n module to every (2m,2n)-tangle. We show that if a link L in S2 x S1 is presented as a closure of a (2n,2n)-tangle, then the Hochschild homology of its H_n bimodule is determined by the link itself and serves as a categorificaiton of J(S2xS1,L). Moreover, we show that this Hochschild homology can be approximated by Khovanov homology of the circular closure of the tangle within S3 by a high twist torus braid, thus providing a practical method of its computation.

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