Virtual tangles and fiber functors

TL;DR

This paper introduces a new category of virtual tangles on surfaces, proving its universality among ribbon categories and extending quantum invariants to virtual links, generalizing classical tangle theory.

Contribution

It defines the category vT of virtual tangles, establishes its universal property among ribbon categories, and extends quantum invariants to virtual links.

Findings

Establishes an equivalence between virtual tangles and a new category vT.

Proves vT's universality among ribbon categories with monoidal functors.

Extends quantum invariants from classical to virtual links.

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.