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

Let $p<p^{\mathrm{\prime }}$ be a pair of coprime positive integers. In this note, generalizing Morton’s work in the case of ${\mathfrak{s}\mathfrak{l}}_2$, we give a formula for the ${\mathfrak{s}\mathfrak{l}}_r$ Jones invariants of torus knots $T(p,p^{\mathrm{\prime }})$ coloured with the finite-dimensional irreducible representations $L_r(n{\mathrm{\Lambda }}_1)$. When $r\le p$, we show that appropriate limits of the shifted (non-normalized, framing dependent) invariants calculated along $L_r(nr{\mathrm{\Lambda }}_1)$ are essentially the characters of certain minimal model principal $\mathcal{W}$ algebras of type $\mathrm{A}$, namely, ${\mathcal{W}}_r(p,p^{\mathrm{\prime }})$, up to some factors independent of $p$ and $p^{\mathrm{\prime }}$ 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 $p<r$.