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

Let $$p<p'$$ be a pair of coprime positive integers. In this note, generalizing Morton’s work in the case $$\mathfrak {sl}_2$$ , we give formula for {sl}_r$$ Jones invariants torus knots $$T (p,p')$$ coloured with finite-dimensional irreducible representations $$L_r(n\Lambda _1)$$ . When $$r \le p$$ show that appropriate limits shifted (non-normalized, framing dependent) calculated along $$L_r(nr\Lambda are essentially characters certain minimal model principal $${\mathcal {W}}$$ algebras type $${{\,\textrm{A}\,}}$$ namely, {W}}_r(p,p')$$ up to some factors independent p and $$p'$$ but depending on r. particular, these modular. We expect 0-tails corresponding sequences invariants. At end, formulate conjecture $$p<r$$