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 $\mathfrak{sl}_2$, we give a formula for the $\mathfrak{sl}_r$ Jones invariants of torus knots $T(p,p')$ coloured with the finite-dimensional irreducible representations $L_r(n\Lambda_1)$. When $r \leq p$, we show that appropriate limits of the shifted (non-normalized, framing dependent) invariants calculated along $L_r(nr\Lambda_1)$ are essentially the characters of certain minimal model principal $\mathcal{W}$ algebras of type $\mathrm{A}$, namely, $\mathcal{W}_r(p,p')$, 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 $p<r$.