On gluing and splitting series invariants of plumbed 3-manifolds

We study series invariants for plumbed 3-manifolds and knot complements twisted by a root lattice. Our series recover recent results of Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park, and Ri and apply more generally to 3-manifolds which are not necessarily negative definite. We show that our series verify certain gluing and splitting properties related to the corresponding operations on 3-manifolds. We conclude with an explicit description of the case of lens spaces and Brieskorn spheres.