XC-tangles and universal invariants

Abstract

We introduce a class of decorated abstract graphs, that we call XC-tangles, that provides a very convenient framework to study quantum invariants of tangles and virtual tangles. These can be viewed as a far-reaching generalisation of rotational tangle diagrams for (virtual) upwards tangles, and constitute the topological analogue of XC-algebras, the minimum algebraic structure needed to construct an knot isotopy invariant following the construction of Lawrence and Lee. XC-tangles admit a very natural description in terms of the so-called XC-Gauss diagrams, and this equivalence lifts the well-known equivalence between virtual upwards tangles and upwards Gauss diagrams. For every XC-algebra $A$, there is a naturally defined strict monoidal full functor $Z_A: \mathcal{T}^{\mathrm{XC}} \rightarrow v\mathcal{E}(A)$ from the category of XC-tangles to the "virtual category of elements of $A$". When $A$ is the endomorphism algebra of a finite-dimensional representation of a ribbon Hopf algebra, this functor can be viewed as an extension of the corresponding Reshetikhin-Turaev functor. Lastly, we also initiate the study of a theory of finite type invariants for one-component XC-tangles that lifts that for virtual long knots.