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

Abstract

We show that the characters of slr versions of the (1, p) singlet and the (1, p) triplet VOAs arise as limits of appropriately coloured slr Jones invariants of certain torus links.

Characters of logarithmic vertex operator algebras and coloured invariants of torus links

“…Our proofs use the formula for these invariants given by Lin and Zheng in [23], and further use combinatorial results to match the limits with appropriate VOAs. Unlike the case of principal W algebras that make appearance in relation to the knots T (p , p) [20], the characters of the singlet and triplet VOAs are either not modular invariant (in the former case) or some modifications are necessary (for the latter case). Their analytic properties have been much studied recently, see for instance [13], [8], [9], [7], etc.…”

“…It is now natural to ask if characters of other VOAs may also similarly arise as appropriate limits of quantum invariants of knots and links. Indeed, in a previous article [20], it was shown that the characters of the principal W-algebras based on sl r for all r ≥ 2 arise as limits of sl r invariants of torus knots. These particular VOAs enjoy very special properties.…”