Virtual tangles and fiber functors

Abstract

We define a category [Formula: see text] 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 [Formula: see text] which induces an equivalence of categories. On the other hand, we show that [Formula: see text] 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.