On Jaeger's HOMFLY-PT expansions, branching rules and link homology: a progress report

TL;DR

This paper discusses generalizing Jaeger's HOMFLY-PT expansion of the Kauffman polynomial to other quantum invariants through branching rules, and explores constructing link homology theories based on these expansions.

Contribution

It introduces a framework for extending Jaeger's HOMFLY-PT expansion to various quantum invariants and proposes a program for developing link homology theories using these expansions.

Findings

Describes Jaeger's HOMFLY-PT expansion of the Kauffman polynomial

Proposes generalization to other quantum invariants via branching rules

Outlines a program to construct link homology theories based on these expansions

Abstract

This note is a write-up of a talk given by the author at the Meeting of the Sociedade Portuguesa de Matematica in July 2012. We describe Jaeger's HOMFLY-PT expansion of the Kauffman polynomial and how to generalize it to other quantum invariants using the so-called "branching rules" for Lie algebra representations. We present a program which aims to construct Jaeger expansions for link homology theories.