Based on previous results of the two first authors, it is shown that the combinatorial construction of invariants of compact, closed three-manifolds by Turaev and Viro as state sums in terms of quantum $6j$-symbols for $\SL_q(2,\C)$ at roots of unity leads to the unitary representation of the mapping class group found by Moore and Seiberg. Via a Heegaard decomposition this invariant may therefore be written as the absolute square of a certain matrix element of a suitable group element in this representation. For an arbitrary Dehn surgery on a figure-eight knot we provide an explicit form for this matrix element involving just one $6j$-symbol. This expression is analyzed numerically and compared with the conjectured large $k=r-2$ asymptotics of the Chern-Simons-Witten state sum [Witten 1989], whose absolute square is the Turaev-Viro state sum. In particular we find numerical agreement concerning the values of the Chern-Simons invariants for the flat $\SU(2)$-connections as predicted by the asymptotic expansion of the state sum with analytical results found by Kirk and Klassen .
"Invariants of three-manifolds, unitary representations of the mapping class group, and numerical calculations." Experiment. Math. 6 (4) 317 - 352, 1997.