Geodesic
Hopf diagrams and quantum invariants
Bruguieres, Alain1 ; Virelizier, Alexis2
1I3M, Université Montpellier II, 34095 Montpellier Cedex 5, France
2Department of Mathematics, University of California, Berkeley CA 94720, USA
Algebraic and Geometric Topology, Tome 5 (2005) no. 4, pp. 1677-1710
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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).

DOI : 10.2140/agt.2005.5.1677
Keywords: Hopf diagrams, string links, quantum invariants

Bruguieres, Alain  1   ; Virelizier, Alexis  2

1 I3M, Université Montpellier II, 34095 Montpellier Cedex 5, France
2 Department of Mathematics, University of California, Berkeley CA 94720, USA 
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Bruguieres, Alain; Virelizier, Alexis. Hopf diagrams and quantum invariants. Algebraic and Geometric Topology, Tome 5 (2005) no. 4, pp. 1677-1710. doi: 10.2140/agt.2005.5.1677

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