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Hopf diagrams and quantum invariants

Alain Bruguieres and Alexis Virelizier

Algebraic & Geometric Topology 5 (2005) 1677–1710

arXiv: math.QA/0505119


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).

Hopf diagrams, string links, quantum invariants
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 18D10, 81R50
Forward citations
Received: 13 June 2005
Accepted: 28 November 2005
Published: 7 December 2005
Alain Bruguieres
Université Montpellier II
34095 Montpellier Cedex 5
Alexis Virelizier
Department of Mathematics
University of California
Berkeley CA 94720