Virtual tangles and fiber functors

Abstract

We define a category vT 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 vT which induces an equivalence of categories. On the other hand, we show that vT 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.

Skein theories for virtual tangles

“…The virtual crossing is neither an overcrossing nor an undercrossing, so one could think of the strands as literally passing through each other. Alternatively, [Bro16] explains that one could imagine this non-planar graph as being embedded on some punctured surface in which the given diagram could be drawn without the need for self-intersection.…”

“…The forbidden move implies that the virtual crossing is not natural with the actual crossing in the planar algebra of unoriented virtual tangles. In tensor category terms, [Bro16] explains that although there is a unique symmetric monoidal functor from the category of tangles to the category of symmetric tangles, that functor is not braided. We explain this phenomenon in planar algebra terms in the following theorem: Theorem 3.2.7 ( [Bro16]).…”

“…In tensor category terms, [Bro16] explains that although there is a unique symmetric monoidal functor from the category of tangles to the category of symmetric tangles, that functor is not braided. We explain this phenomenon in planar algebra terms in the following theorem: Theorem 3.2.7 ( [Bro16]). Let T virt be the planar algebra of unoriented virtual tangles, P be a symmetric planar algebra, and A be a braided sub-planar algebra of P. Then there exists a canonical map from T virt → P 3.2.1.…”

Circuit algebras are wheeled props

“…Virtual tangles -a generalization of classical tangles -were the motivating example for the definition of circuit algebras, and so far they have been their main area of application [Bro19,Pol10]. An oriented (classical) tangle is a smooth embedding of an oriented 1-manifold M into a 3-dimensional ball M → B 3 , such that the boundary is mapped to the boundary of the ball: ∂M → S 2 = δB 3 .…”

Representations of the loop braid group and Aharonov–Bohm like effects in discrete $(3+1)$-dimensional higher gauge theory

“…Suppose we have a monoidal representation of B in a symmetric category; the main example to have in mind is the category of vector spaces. This lifts automatically [25] to a representation of VB by mapping V 1 [2] to the transposition of tensor factors. However, for representations of the loop braid group we require the welded braid group, and in this the welded move (6) is also satisfied.…”