## Experimental Mathematics

- Experiment. Math.
- Volume 6, Issue 4 (1997), 317-352.

### Invariants of three-manifolds, unitary representations of the mapping class group, and numerical calculations

Michael Karowski, Robert Schrader, and Elmar Vogt

#### 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 $\SL_q(2,\C)$ 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].

#### Article information

**Source**

Experiment. Math., Volume 6, Issue 4 (1997), 317-352.

**Dates**

First available in Project Euclid: 7 March 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1047047192

**Mathematical Reviews number (MathSciNet)**

MR1606924

**Zentralblatt MATH identifier**

1115.57300

**Subjects**

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]

Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]

#### Citation

Karowski, Michael; Schrader, Robert; Vogt, Elmar. Invariants of three-manifolds, unitary representations of the mapping class group, and numerical calculations. Experiment. Math. 6 (1997), no. 4, 317--352. https://projecteuclid.org/euclid.em/1047047192