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?