Coloured Invariants of Torus Knots, Algebras, and Relative Asymptotic Weight Multiplicities

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 \(p,p'\ge h^\vee \), we use the representation theory of the corresponding principal affine 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 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.