Higher Rank (Z)over-cap and FK

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 G = SU(2) case. Although a full mathematical definition for these "invariants" is lacking yet, we define (Z) over cap (G) for negative definite plumbed 3-manifolds and F-K(G) for torus knot complements. As in the G = SU(2) case by Gukov and Manolescu, there is a surgery formula relating F-K(G) to (Z) over cap (G) of a Dehn surgery on the knot K. Furthermore, specializing to symmetric representations, F-K(G) 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.