Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots

TL;DR

This paper develops a universal framework for adjoint knot polynomials using Racah matrices, unifying descriptions across various Lie algebras and knot types, and extends the eigenvalue conjecture to 6x6 matrices.

Contribution

It extends the universality from quantum dimensions to Racah matrices, providing a unified approach to adjoint knot polynomials for all arborescent knots and advancing the eigenvalue conjecture.

Findings

Unified description of adjoint knot polynomials for arborescent knots.

Extended eigenvalue conjecture to 6x6 Racah matrices.

Demonstrated universality across different Lie algebras and knot families.

Abstract

By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY (SU_N) and Kauffman (SO_N) polynomials. For E_8 the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot polynomials for all arborescent (double-fat) knots, including twist, 2-bridge and pretzel. Technically we develop together the universality and the "eigenvalue conjecture", which expresses the Racah and mixing matrices through the eigenvalues of the quantum R-matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown 6x6 case. The adjoint polynomials do not distinguish between mutants and therefore are not very efficient in knot theory, however, universal polynomials in higher representations can probably be better in this respect.