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

Abstract

Let $p<p^{\prime }$ 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 $T(p,p^{\prime })$ coloured with the finite-dimensional irreducible representations $L_r(n{\Lambda }_1)$. When $r\le p$, 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, ${\mathcal{W}}_r(p,p^{\prime })$, up to some factors independent of $p$ and $p^{\prime }$ 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 $p<r$.

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References in corpus (2)

• Difference equation of the colored Jones polynomial for torus knot
• A $q$-series identity via the ${\mathfrak{sl}}_3$ colored Jones polynomials for the $(2,2m)$-torus link