Polynomial link invariants and quantum algebras

TL;DR

This paper explores the algebraic structures underlying quantum link invariants, such as the Jones and HOMFLY-PT polynomials, providing a unified framework that extends these invariants through quantum algebras.

Contribution

It introduces an algebraic setup based on quantum Schur--Weyl duality that unifies the definitions of various quantum link invariants, including the HOMFLY-PT polynomial.

Findings

Unified algebraic framework for quantum link invariants

Extensions of quantum objects to encompass all classical invariants

Connections between quantum groups and classical polynomial invariants

Abstract

The definition of the Jones polynomial in the 80's gave rise to a large family of so-called quantum link invariants, based on quantum groups. These quantum invariants are all controlled by the same two-variable invariant (the HOMFLY-PT polynomial), which also specializes to the older Alexander polynomial. Building upon quantum Schur--Weyl duality and variants of this phenomenon, I will explain an algebraic setup that allows for global definitions of these quantum polynomials, and discuss extensions of these quantum objects designed to encompass all of the mentioned invariants, including the HOMFLY-PT polynomial.