Webs and Clasps

Abstract: The discovery of the Jones polynomial triggered mathematical developments in areas including knot theory and quantum algebra. One way to define the Jones polynomial is by using the braiding in the Temperley-Lieb category, which can be defined with planar matching. We can use diagrams and graphical calculations in the Temperley-Lieb category to study the rep- resentation theory of quantum sl2. The irreducible representations can be “visualized” as the Jones-Wenzl projectors, which can be used to compute colored Jones polynomial and quantum sl2 3-manifold invariant.

The sl2 case is generalized to other simple Lie algebras by introducing triva- lent vertices, and the generalized graphical categories are called spiders or web categories. Clasps are defined as analogues of the Jones-Wenzl projectors, and we can use clasps to compute colored quantum link invariants, quantum 3- manifold invariants, 3-j symbols, and 6-j symbols of different quantum groups.

In this talk, I will review the background material, and talk about re- cent developments on definition of web categories and clasp expansions for different Lie types.

algebraic topologycategory theorygroup theoryK-theory and homology

Audience: researchers in the topic

Bilkent Topology Seminar

Series comments: Contact the organizer to get access to Zoom.

Recordings of talks available at www.youtube.com/channel/UCLrmyGpqxyeVpTcA1b5HcMw/videos