Torus knots in adjoint representation and Vogel's universality

TL;DR

This paper explores Vogel's universality in Lie algebra representation theory, extending universal knot invariants to include adjoint invariants for torus knots, especially focusing on T[4,n] with odd n.

Contribution

It extends the list of universal knot invariants within Vogel's framework and details the computation of adjoint invariants for specific torus knots.

Findings

Extended universal knot invariants to more Lie algebras.

Derived explicit formulas for T[4,n] torus knots with odd n.

Connected Vogel's parameters to knot invariants.

Abstract

Vogel's universality gives a unified description of the adjoint sector of representation theory for simple Lie algebras in terms of three parameters $\alpha,\beta,\gamma$, which are homogeneous coordinates of Vogel's plane. It is associated with representation theory within the framework of Chern-Simons theory only, and gives rise to universal knot invariants. We extend the list of these latter further, and explain how to deal with the adjoint invariants for the torus knots $T[m,n]$ considering the case of $T[4,n]$ with odd $n$ in detail.