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Hopf diagrams and quantum invariants. (English) Zbl 1116.57011
The paper discusses certain aspects of an abstract general construction of quantum manifold invariants. Its main goal is to introduce a monoidal category of Hopf diagrams and to describe a universal encoding of ribbon strings as Hopf diagrams. By definition, a Hopf diagram is a morphism of the category Diag, which is the convolution category \(\text{Conv }_{\mathcal D}(*,{\mathbf 1})\) of the category \({\mathcal D}\) generated by one object \(*\) and the morphisms \(\Delta :*\rightarrow *\otimes *,\epsilon :*\rightarrow {\mathbf 1}, \omega_+:*\otimes *\rightarrow {\mathbf 1}, \omega_{-}:*\otimes *\rightarrow {\mathbf 1}, \theta _+:*\rightarrow {\mathbf 1}, \theta_{-}:*\rightarrow {\mathbf 1},\) where \({\mathbf 1}\) denotes the unit object of the monoidal category, subject to the relations \((\text{id}_*\otimes \Delta )\Delta=(\Delta \otimes \text{id}_*)\Delta\) and \((\text{id}_*\otimes \epsilon )\Delta=\text{id}_*=(\epsilon \otimes \text{id}_*)\Delta\). The universal encoding introduced by the authors is an injective monoidal functor and admits a straightforward monoidal retraction. Any Hopf diagram with \(n\) legs yields an \(n\)-form on the coend of a ribbon category in a completely explicit way. So the computation of a quantum invariant of a \(3\)-manifold is reduced to a two stage process: first the formal computation of the associated Hopf diagram, and second, the evaluation of this diagram in the given category. The latter relies on the so-called Kirby elements, which are also discussed in the paper.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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