Research PaperNo Access

Virtual tangles and fiber functors

Adrien Brochier

Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

https://doi.org/10.1142/S0218216519500445Cited by:1 (Source: Crossref)

Abstract

We define a category $v 𝒯$ of tangles diagrams drawn on surfaces with boundaries. On the one hand, we show that there is a natural functor from the category of virtual tangles to $v 𝒯$ which induces an equivalence of categories. On the other hand, we show that $v 𝒯$ is universal among ribbon categories equipped with a strong monoidal functor to a symmetric monoidal category. This is a generalization of the Shum–Reshetikhin–Turaev theorem characterizing the category of ordinary tangles as the free ribbon category. This gives a straightforward proof that all quantum invariants of links extend to framed oriented virtual links. This also provides a clear explanation of the relation between virtual tangles and Etingof–Kazhdan formalism suggested by Bar-Natan. We prove a similar statement for virtual braids, and discuss the relation between our category and knotted trivalent graphs.

Keywords:

AMSC: 17B37, 57M27

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