Higher Rank Ẑ and FK

Abstract

We study q-series-valued invariants of 3-manifolds that depend on the choice of a root system 𝐺. This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057], where they focused on the 𝐺 = SU(2) case. Although a full mathematical definition for these ''invariants'' is lacking yet, we define Ẑ𝐺 for negative definite plumbed 3-manifolds and FGK for torus knot complements. As in the 𝐺 = SU(2) case by Gukov and Manolescu, there is a surgery formula relating FGK to Ẑ𝐺 of a Dehn surgery on the knot K. Furthermore, specializing to symmetric representations, FGK satisfies a recurrence relation given by the quantum A-polynomial for symmetric representations, which hints that there might be HOMFLY-PT analogues of these 3-manifold invariants.