Coloured $\mathfrak{sl}_r$ invariants of torus knots and characters of $\mathcal{W}_r$ algebras

Abstract

Let p < p ′ be a pair of coprime positive integers. In this note, generalizing Morton's work in the case of sl2, we give a formula for the slr Jones invariants of torus knots T (p, p ′ ) coloured with the finite-dimensional irreducible representations Lr(nΛ1). When r ≤ p, we show that appropriate limits of the shifted (non-normalized, framing dependent) invariants calculated along Lr(nrΛ1) are essentially the characters of certain minimal model principal W algebras of type A, namely, Wr(p, p ′ ), up to some factors independent of p and p ′ but depending on r. In particular, these limits are essentially modular. We expect these limits to be the 0-tails of corresponding sequences of invariants. At the end, we formulate a conjecture on limits for p < r.