Algebraic and Geometric Topology 5 (2005), paper no. 68, pages 1677-1710.

Hopf diagrams and quantum invariants

Alain Bruguières and Alexis Virelizier

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

AMS subject classification. Primary: 57M27. Secondary: 18D10, 81R50.

E-print: arXiv:math.QA/0505119

DOI: 10.2140/agt.2005.5.1677

Submitted: 13 June 2005. Accepted: 28 November 2005. Published: 7 December 2005.

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Alain Bruguières and Alexis Virelizier
I3M, Universite Montpellier II, 34095 Montpellier Cedex 5, France
Department of Mathematics, University of California, Berkeley CA 94720, USA

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