Coloured invariants of torus knots, $\mathcal{W}$ algebras, and relative asymptotic weight multiplicities

We study coloured invariants of torus knots $T(p,p^{\mathrm{\prime }})$ (where $p,p^{\mathrm{\prime }}$ are coprime positive integers). When the colouring Lie algebra is simply-laced, and when $p,p^{\mathrm{\prime }}\ge h^{\vee }$, we use the representation theory of the corresponding principal affine $\mathcal{W}$ algebras to understand the trailing monomials of the coloured invariants. In these cases, we show that the appropriate limits of the renormalized invariants are equal to the characters of certain $\mathcal{W}$ algebra modules (up to some factors). This result on limits rests on a purely Lie-algebraic conjecture on asymptotic weight multiplicities which we verify in some examples.