An early highlight of quantum topology was the observation that the Jones polynomial — and many other knot and link invariants — arise from braided monoidal categories of quantum group representations. In hindsight, this can be understood as underlying reason for the existence  of associated topological quantum field theories (TQFTs) in 3 and 4 dimensions.

Not much later, Khovanov discovered a link homology theory that categorifies the Jones polynomial. It associates graded chain complexes to links, from which the Jones polynomials can be recovered. It was therefore speculated that Khovanov homology and its variants may themselves be expressible in terms of certain braided monoidal 2-categories and that there should exist associated TQFTs in 4 and 5 dimensions that may be sensitive to smooth structure.

A major challenge in fully realizing this dream is the problem of coherence: Link homology theories live in the world of homological algebra, where constructing a braided monoidal structure in principle requires an infinite amount of higher and higher homological coherence data. In this talk, I will sketch a proposed solution to this problem, joint with Leon Liu, Aaron Mazel-Gee, David Reutter, and Catharina Stroppel, and explain how we use the language of infinity-categories to build an E2-monoidal (∞,2)-category which categorifies the Hecke braided monoidal category underlying the HOMFLYPT link polynomial.