  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.
