Jennifer Brown, "Defects in Skein Theory and the A-polynomial"
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Europe/Paris
Description
Skein theory is a tool for studying categories through graphical calculus. When the input is an appropriate category of representations, it provides a deformation quantization of a moduli space of local systems on a 3-manifold. This combination of representation theory, geometry, and topology leads to a rich family of algebras and modules.
In the first part of the talk we will introduce the graphical calculus of ribbon categories, then use it to define skein relations, algebras, and modules. We will then explain how these constructions are modified to allow for interfaces (or "defects") between different ribbon categories.
Our main motivation for introducing defects will be ongoing work to quantize the A-polynomial. This is a two-variable knot polynomial constructed from the space of SL2C local systems on the knot complement.
In the second part we will define the A-polynomial and show how defect skein theory appears naturally in it's quantization.