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2005 Hopf diagrams and quantum invariants
Alain Bruguieres, Alexis Virelizier
Algebr. Geom. Topol. 5(4): 1677-1710 (2005). DOI: 10.2140/agt.2005.5.1677

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).

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Alain Bruguieres. Alexis Virelizier. "Hopf diagrams and quantum invariants." Algebr. Geom. Topol. 5 (4) 1677 - 1710, 2005. https://doi.org/10.2140/agt.2005.5.1677

Information

Received: 13 June 2005; Accepted: 28 November 2005; Published: 2005
First available in Project Euclid: 20 December 2017

zbMATH: 1116.57011
MathSciNet: MR2186115
Digital Object Identifier: 10.2140/agt.2005.5.1677

Subjects:
Primary: 57M27
Secondary: 18D10, 81R50

Rights: Copyright © 2005 Mathematical Sciences Publishers

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