Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalism

TL;DR

This paper provides a direct proof that the normalized colored Alexander polynomial for one-hook representations exhibits a specific scaling property, using Reshetikhin-Turaev formalism to derive R-matrices.

Contribution

It offers a simple, direct proof of the one-hook scaling property for the Alexander polynomial via Reshetikhin-Turaev formalism, clarifying its mathematical structure.

Findings

Proved the scaling property for one-hook representations.

Used Reshetikhin-Turaev formalism to derive R-matrices.

Established a direct link between colored and fundamental Alexander polynomials.

Abstract

We prove that normalized colored Alexander polynomial (the $A \rightarrow 1$ limit of colored HOMFLY-PT polynomial) evaluated for one-hook (L-shape) representation R possesses scaling property: it is equal to the fundamental Alexander polynomial with the substitution $q \rightarrow q^{|R|}$. The proof is simple and direct use of Reshetikhin-Turaev formalism to get all required R-matrices.