Abstract
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 SLq(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 statesum with analytical results found by Kirk and Klassen [1990].