Eigenvalue conjecture and colored Alexander polynomials

TL;DR

This paper explores the connection between two major conjectures in knot polynomial theory, providing new support for the eigenvalue conjecture through properties of Alexander polynomials and their relation to mixing matrices.

Contribution

It links the property of Alexander polynomials for single hook Young diagrams to the eigenvalue conjecture, offering indirect evidence for the universality of mixing matrices in knot theory.

Findings

Supports the eigenvalue conjecture via Alexander polynomial properties

Provides a new perspective on mixing matrices in braid group representations

Strengthens the evidence for the eigenvalue conjecture for complex cases

Abstract

We connect two important conjectures in the theory of knot polynomials. The first one is the property Al_R(q) = Al_{[1]}(q^{|R|}) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices U_{i} in the relation {\cal R}_i = U_i{\cal R}_1U_i^{-1} between the i-th and the first generators {\cal R}_i of the braid group are universally expressible through the eigenvalues of {\cal R}_1. Since the above property of Alexander polynomials is very well tested, this relation provides a new support to the eigenvalue conjecture, especially for i>2, when its direct check by evaluation of the Racah matrices and their convolutions is technically difficult.