Categorifications of affine tangles
Rina Anno, Harvard

A categorification, or rather a functor-valued invariant, of a particular kind of tangles, is an assignment of a category to each boundary piece of a tangle, and to each tangle, a functor between the corresponding categories. Various categorifications of oriented tangles have been developed recently, mainly in the course of obtaining invariants of links. We are going to consider another side of the story, namely, how the existence of a tangle representation helps to understand the structure of the underlying categories. For affine tangles (tangles in an annulus) there are several examples where the categories may be described in combinatorial terms of crossingless matchings of 2n points on a circle and modules over tensor powers of the Frobenius algebra $A = C[X]/X^2$ of dual numbers. Thus, unlike the linear case, affine tangles provide a series of non-isomorphic categorifications indexed by A-modules.