Hopf diagrams and quantum invariants

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).