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2-tangles. (English) Zbl 0905.57017

In the note under review, by semistrict braided monoidal \(2\)-categories are meant those defined by J. C. Baez and M. Neuchl [Adv. Math. 121, No. 2, 196-244 (1996; Zbl 0855.18008)], but satisfying certain extra axioms introduced by S. Crans [Generalized centers of braided and sylleptic monoidal \(2\)-categories. To appear in Adv. Math.]. The note is in fact an announcement of a theorem claiming that the \(2\)-category of unframed unoriented \(2\)-tangles in four dimensions can be characterized as the “free semistrict braided monoidal \(2\)-category with duals on one unframed self-dual object”. A proof using the movie moves of Carter, Rieger, and Saito is promised to be given in a forthcoming paper.
The authors have hopes that there are interesting examples of semistrict braided monoidal \(2\)-categories. They comment on how the above mentioned characterization might be used to construct invariants of \(2\)-tangles in four dimensions.

MSC:

57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
57R40 Embeddings in differential topology

Citations:

Zbl 0855.18008
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