One rational expression in Vogel parameters computes quantum dimensions of Cartan powers of adjoint and X2 representations for many simple Lie algebras, though it vanishes in some cases where the representation exists.
$X_2$ series of universal quantum dimensions
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abstract
The antisymmetric square of the adjoint representation of any simple Lie algebra is equal to the sum of adjoint and $X_2$ representations. We present universal formulae for quantum dimensions of an arbitrary Cartan power of $X_2$. They are analyzed for singular cases and permuted universal Vogel's parameters. $X_2$ has been the only representation in the decomposition of the square of the adjoint with unknown universal series. Application to universal knot polynomials is discussed.
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On universal quantum dimensions of certain two-parameter series of representations
One rational expression in Vogel parameters computes quantum dimensions of Cartan powers of adjoint and X2 representations for many simple Lie algebras, though it vanishes in some cases where the representation exists.