Quantum dimensions of adjoint representations and of their descendants, as well as the associated Racah matrices are described by universal 3-parametric formulas, independent of the simple Lie algebra.

This allows to define "universal knot polynomials" in the "$E_8$-sector" of representation theory, see arXiv:1510.05884 and arXiv:1511.09077

To obtain adjoint-colored polynomials one puts

$u=q^{-2}, \ v=q^2, \ w=q^N=A\ $ for $SU(N)$ ("uniform" HOMFLY)

$u=q^{-2}, \ v=q^4, \ w=q^{N-4}=A/q^3\ $ for $SO(N)$ (adjoint Kauffman)

$u=q^{-2},\ v = q, \ w=q^{N/2+2}\ $ for $Sp(N)$

$u=q^{-2},\ v=q^{N+4}, \ w=q^{2N+4}\ $ for exceptional line with $\ N_{G_2}=-2/3,\ N_{F_4}=1,\ N_{E_6}=2,\ N_{E_7} = 4,\ N_{E_8}=8$.

- Universal adjoint polynomials
- Some universal dimensions and Racah matrix for $Adj^{\otimes 3}\longrightarrow Adj$