A state-sum formula for the Alexander polynomial

TL;DR

This paper introduces a new diagrammatic state-sum approach to compute the Alexander polynomial of braid closures, utilizing Markov trace formulas and Young's semi-normal representations of Iwahori-Hecke algebras.

Contribution

It presents a novel diagrammatic formalism for the Alexander polynomial based on advanced algebraic tools, bridging braid theory and algebraic representations.

Findings

Derived a state-sum formula for the Alexander polynomial

Connected Markov trace formulas with Young's semi-normal representations

Provided a computational framework for braid closures

Abstract

We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori-Hecke algebras of type A.