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Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 216
Invariants of 3-manifolds, Unitary representations of the mappingclass group and numerical calculations M. Karowski, R. Schrader
Based on previous results of the two first authors, it is shown that the combinatorial construction of invariants of compact, closed 3-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 asymptotics of the Chern-Simons-Witten state sum, 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.
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