Categorification: tangle invariants and TQFTs

Abstract

Based on different views on the Jones polynomial, we review representation theoretic categorified link and tangle invariants. We unify them in a common combinatorial framework and connect them via the theory of Soergel bimodules. The influence of these categorifications on the development of 2-representation theory and the interaction between topological invariants and 2-categorical structures is discussed. Finally, we indicate how categorified representations of quantum groups, on the one hand, and monoidal 2-categories of Soergel bimodules, on the other hand, might lead to new interesting 4-dimensional TQFTs.