Eigenvalue conjecture and colored Alexander polynomials

A Mironov,A Morozov

doi:10.1140/epjc/s10052-018-5765-5

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 mathcal{R}_i = U_imathcal{R}_1U_i^{-1} between the ith and the first generators mathcal{R}_i of the braid group are universally expressible through the eigenvalues of mathcal{R}_1. Since the above property of Alexander polynomials is very well tested, this relation provides 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.

Highlights

An indisputable advantage of knot theory from the point of view of representation theory is that the former provides a set of quantities that adequately capture and reveal the basic hidden properties of the latter
The power of knot polynomial quantum field theory (QFT) methods in knot and representation theories is an impressive manifestation of the effectiveness of string theory approach to mathematical problems, especially when their calculational aspects are concerned
Equation (2), got less attention and still remains a mystery. We reduce it to another “experimental” discovery, the eigenvalue conjecture of [18], (2) follows from its stronger version, applicable to arbitrary number m of strands in the braid

Summary

Introduction

An indisputable advantage of knot theory from the point of view of representation theory is that the former provides a set of quantities that adequately capture and reveal the basic hidden properties of the latter. For all knots K and all single hook Young diagrams R = [r, 1s] of size |R| = r + s.

Results

Conclusion