  We study coloured invariants of torus knots $T(p,p')$ (where $p,p'$ are
coprime positive integers). When the colouring Lie algebra is simply-laced, and
when $p,p'\geq h^\vee$, 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.
