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

m a t h f r a k {s l}_{2}

, we give a formula for the

m a t h f r a k {s l}_{r}

Jones invariants of torus knots

T (p, p^{'})

coloured with the finite-dimensional irreducible representations

L_{r} (n L a m b d a_{1})

. When

r l e q p

, we show that appropriate limits of the shifted (non-normalized, framing dependent) invariants calculated along

L_{r} (n r L a m b d a_{1})

are essentially the characters of certain minimal model principal

m a t h c a l W

algebras of type

m a t h r m A

, namely,

m a t h c a l 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

.

This page was built for publication: Coloured $\mathfrak{sl}_r$ invariants of torus knots and characters of $\mathcal{W}_r$ algebras