Torus knots in adjoint representation and Vogel’s universality

Abstract Vogel’s universality gives a unified description of the adjoint sector of representation theory for simple Lie algebras in terms of three parameters $$\alpha ,\beta ,\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> , 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.