The Reshetikhin–Turaev invariant, Turaev’s TQFT, and many related constructions rely on the encoding of certain tangles (–string links, or ribbon –handles) as –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 legs yields a –form on the coend of a ribbon category in a completely explicit way. Thus computing a quantum invariant of a –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).
"Hopf diagrams and quantum invariants." Algebr. Geom. Topol. 5 (4) 1677 - 1710, 2005. https://doi.org/10.2140/agt.2005.5.1677