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Integrality of quantum 3-manifold invariants and a rational surgery formula

Part of: Topological manifolds Low-dimensional topology

Published online by Cambridge University Press:  01 November 2007

Anna Beliakova  and
Thang T. Q. Lê
Anna Beliakova
Affiliation:
Institut für Mathematik, Universität Zurich, Winterthurerstrasse 190, 8057 Zürich, Switzerland (email: anna@math.unizh.ch)
Thang T. Q. Lê
Affiliation:
Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA (email: letu@math.gatech.edu)
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Abstract

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We prove that the Witten–Reshetikhin–Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an application, we compute the unified invariant for Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this invariant separates Seifert fibered integral homology spaces and can be used to detect the unknot.

Keywords

3-manifoldsunified quantum invariantscyclotomic completion ringq-hypergeometric series

MSC classification

Secondary: 57N10: Topology of general $3$-manifolds 57M25: Knots and links in $S^3$
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 6 , November 2007 , pp. 1593 - 1612
Copyright
Copyright © Foundation Compositio Mathematica 2007