Topology Seminar: Towards knot homology for 3-manifolds

Monday, November 15, 2021 at 2:05pm to 3:05pm

This is a past event.

Kaprielian Hall (KAP), 245
3620 South Vermont Avenue, Los Angeles, CA 90089


Aaron Mazel-Gee, Caltech

Abstract: The Jones polynomial is an invariant of knots in R^3. Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin--Turaev using quantum groups.

Khovanov homology is a categorification of the Jones polynomial of a knot in R^3, analogously to how ordinary homology is a categorification of the Euler characteristic of a space. It is a major open problem to extend Khovanov homology to knots in 3-manifolds.

In this talk, I will explain forthcoming work towards solving this problem, joint with Leon Liu and David Reutter. Roughly speaking, our contribution amounts to the first instance of a braiding on 2-representations of a categorified quantum group. More precisely, we construct a braided (∞,2)-category that simultaneously incorporates all of Rouquier's braid group actions on Hecke categories in type A, articulating a novel compatibility among them.

 

Event Type

Lecture / Talk / Workshop

Campus

University Park Campus

Department
Mathematics
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