Geometry & Topology Monographs, Vol. 4 (2002),
Invariants of knots and 3-manifolds (Kyoto 2001),
Paper no. 6, pages 69--87.

Quantum invariants of Seifert 3-manifolds and their asymptotic expansions

Soren Kold Hansen Toshie Takata


Abstract. We report on recent results of the authors concerning calculations of quantum invariants of Seifert 3-manifolds. These results include a derivation of the Reshetikhin-Turaev invariants of all oriented Seifert manifolds associated with an arbitrary complex finite dimensional simple Lie algebra, and a determination of the asymptotic expansions of these invariants for lens spaces. Our results are in agreement with the asymptotic expansion conjecture due to JE Andersen [The Witten invariant of finite order mapping tori I, to appear in J. Reine Angew. Math.] and [The asymptotic expansion conjecture, from `Problems on invariants of knots and $3$--manifolds', edited by T. Ohtsuki, http://www.ms.u-tokyo.ac.jp/~tomotada/proj01/].

Keywords. Quantum invariants, Seifert manifolds, modular categories, quantum groups, asymptotic expansions

AMS subject classification. Primary: 57M27. Secondary: 17B37, 18D10, 41A60.

E-print: arXiv:math.GT/0210011

Submitted to GT on 3 December 2001. Paper accepted 22 July 2002. Paper published 19 September 2002.

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Soren Kold Hansen Toshie Takata
Dept of Maths and Stats, University of Edinburgh, JCMB
King's Buildings, Edinburgh EH9 3JZ, UK

Department of Mathematics, Faculty of Science
Niigata University, Niigata 950-2181, Japan

Email: hansen@maths.ed.ac.uk, takata@math.sc.niigata-u.ac.jp

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