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.
Notes on file formats
Alain Bruguières and Alexis Virelizier
I3M, Universite Montpellier II, 34095 Montpellier Cedex 5, France
and
Department of Mathematics, University of California, Berkeley CA 94720, USA
Email: bruguier@math.univ-montp2.fr, virelizi@math.berkeley.edu
AGT home page
