Algebraic & Geometric Topology

Hopf diagrams and quantum invariants

Alain Bruguieres and Alexis Virelizier

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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).

Article information

Algebr. Geom. Topol., Volume 5, Number 4 (2005), 1677-1710.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]

Hopf diagrams string links quantum invariants


Bruguieres, Alain; Virelizier, Alexis. Hopf diagrams and quantum invariants. Algebr. Geom. Topol. 5 (2005), no. 4, 1677--1710. doi:10.2140/agt.2005.5.1677.

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