ar X iv : m at h / 05 05 11 9 v 1 [ m at h . Q A ] 6 M ay 2 00 5 HOPF DIAGRAMS AND QUANTUM INVARIANTS

The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain ribbon tangles (n-string links, or n-ribbon 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 retrac-tion. 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.