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Journal Article

Virtual tangles and fiber functors

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Brochier,  Adrien
Max Planck Institute for Mathematics, Max Planck Society;

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arXiv:1602.03080.pdf
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Citation

Brochier, A. (2019). Virtual tangles and fiber functors. Journal of Knot Theory and Its Ramifications, 28(7): 1950044. doi:10.1142/S0218216519500445.


Cite as: https://hdl.handle.net/21.11116/0000-0004-8AD1-4
Abstract
We define a category $v\mathcal{T}$ 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 $v\mathcal{T}$ which induces an equivalence of categories. On the other hand, we show that $v\mathcal{T}$ 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 extends 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.