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 [1990].