Hopf diagrams and quantum invariants

Bruguieres, Alain; Virelizier, Alexis

doi:10.2140/agt.2005.5.1677

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Hopf diagrams and quantum invariants

Alain Bruguieres and Alexis Virelizier

Algebraic & Geometric Topology 5 (2005) 1677–1710

DOI: 10.2140/agt.2005.5.1677

arXiv: math.QA/0505119

Abstract

The Reshetikhin–Turaev invariant, Turaev’s TQFT, and many related constructions rely on the encoding of certain tangles ( $n$ –string links, or ribbon $n$ –handles) as $n$ –forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hopf diagrams. This universal encoding is an injective monoidal functor and admits a straightforward monoidal retraction. Any Hopf diagram with $n$ legs yields a $n$ –form on the coend of a ribbon category in a completely explicit way. Thus computing a quantum invariant of a $3$ –manifold reduces to the purely formal computation of the associated Hopf diagram, followed by the evaluation of this diagram in a given category (using in particular the so-called Kirby elements).

Keywords

Hopf diagrams, string links, quantum invariants

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 18D10, 81R50

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Publication

Received: 13 June 2005

Accepted: 28 November 2005

Published: 7 December 2005

Authors

Alain Bruguieres
I3M Université Montpellier II 34095 Montpellier Cedex 5 France

Alexis Virelizier
Department of Mathematics University of California Berkeley CA 94720 USA

Volume 5, issue 4 (2005)