On the computation of the Turaev-Viro module of a knot

Let M be the manifold obtained by 0-framed surgery along a knot K in the 3-sphere. A Topological Quantum Field Theory assigns to a fundamental domain of the universal abelian cover of M an operator, whose non-nilpotent part is the Turaev-Viro module of K. In this paper, using surgery formulas, we give a matrix presentation for the Turaev-Viro module of any knot Ii, in the case of the (V-p, Z(p)) TQFT of Blanchet, Habegger, Masbaum and Vogel. We do the computation for a family of knots in the special case p = 8, and note the relation with the fibering question.