Matrix integral expansion of colored Jones polynomials for figure-eight knot

TL;DR

This paper explores extending matrix integral representations to non-torus knots, specifically analyzing SU(2) Racah coefficients via double matrix integrals and uncovering their algebraic properties.

Contribution

It introduces a novel approach to represent SU(2) Racah coefficients as double matrix integrals, expanding the scope of matrix integral methods in knot theory.

Findings

Racah coefficients are expressed as expansion coefficients in double matrix integrals.

The transformed coefficients exhibit notable algebraic properties.

Potential extension of matrix integral representations beyond torus knots.

Abstract

In this note we examine a possible extension of the matrix integral representation of knot invariants beyond the class of torus knots. In particular, we study a representation of the SU(2) quantum Racah coefficients by double matrix integrals. We find that the Racah coefficients are mapped to expansion coefficients in some basis of double integrals. The transformed coefficients have a number of interesting algebraic properties.