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First, I would like to know how many definitions are there for categorification of WRT invariants. In addition, I wonder if the categorified version of WRT invariants have been explicitly computed for some integral homology spheres.

In the case of ordinary WRT invariants, sl(2) invariants can be expressed by the summation of colored Jones polynomials with the modular S-matrices over all the colors. Could the categorified invariants for integral homology spheres be computable if the colored sl(2)-homology, say, of torus knots or twist knots, are known? If so, what replaces the modular S-matrices?

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    $\begingroup$ Last I knew, there were no rigorous mathematical constructions of categorified WRT invariants of 3 manifolds. Of course, lots of people would like to fix this, but at the moment, I don't think anyone knows, say, what replaces S-matrices. $\endgroup$
    – Ben Webster
    Apr 2, 2013 at 16:50
  • $\begingroup$ Thanks for your comments, Ben. Are the expressions for categorified invariants expected to have all positive integer coefficients? In the paper by Aganagic and Shakirov (arxiv.org/abs/1105.5117), they propose the refined modular S-matrix which is written $M_{R_i}(t^{\rho})M_{R_j}(t^{\rho}q^{R_i})$ where $M_{R}(x_1,\cdots,x_N;q,t)$ is the Macdonald polynomial with highest weight $R$. Naively thinking, this is a nice candidate for what replaces the modular S-matrix. $\endgroup$ Apr 2, 2013 at 17:28

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