Uniqueness and branching of knot homologies More Info

Knot homology theories are categorified versions of classical (and quantum) knot polynomials that are (conjecturally/almost) functorial under knot cobordisms. The goal of this talk is to present two examples of the role of higher representation theory in the study of categorifications of the Reshetikhin-Turaev invariants of knots colored with 𝔰𝔩(N) representations. As a first example, I will explain how categorical versions of skew Howe duality have been used to prove that several, superficially very different constructions of 𝔰𝔩(N) knot homologies (via matrix factorizations, category , coherent sheaves...) produce isomorphic invariants. The second example concerns categorified branching rules, which provide relationships between these knot homologies in the form of spectral sequences, which are also interesting from a topological perspective. If time permits, I will talk about some open problems related to colored HOMFLY-PT homologies, whose underlying higher representation theory is less well understood.