Mathematical Physics Seminar: Adrien Brochier (Paris Diderot)

Mathematical Physics Seminar, external speaker

1h virtual talk in MS Teams, followed by informal discussion

Speaker: Adrien Brochier (Paris Diderot)

Title: Quantum invariants and virtual knots

Abstract: Shum--Reshetikhin--Turaev theorem characterizes the category of framed oriented tangles as the free ribbon category. This is the basic ingredient underlying the construction of knots invariants coming from quantum groups, and the closely related combinatorial version of the Kontsevich integral. In turn, the latter construction has a natural interpretation in the realm of deformation quantization, and explains to some extent the very existence of the ribbon category of representations of quantum groups, but not quite the existence of quantum groups themselves.
I will overview this part of the theory, and then talk about a generalization of Shum--Reshetikhin--Turaev theorem to virtual tangles. Roughly speaking, I'll explain that the category of framed orientd virtual tangles is universal among ribbon categories equipped with a fiber functor. This shows that invariant of knots coming from quantum groups (or more generally ribbon Hopf algebras, as opposed to general ribbon categories) extends to virtual knots. This is motivated by "Bar-Natan's dream" that a still hypothetical Kontsevich integral for virtual knots should explain Etingof--Kazhdan quantization of Lie bialgebras.