Z and Splice Diagrams

We study quantum q-series invariants of 3-manifolds Z_σ of Gukov-Pei-Putrov-Vafa, using techniques from the theory of normal surface singularities such as splice diagrams. We show that the (suitably normalized) sum of all Z_σ depends only on the splice diagram, and in particular, it agrees for manifolds with the same universal abelian cover. We use these ideas to find simple formulas for Z_σ invariants of Seifert manifolds. Applications include a better understanding of the vanishing of the q-series Z_σ. Additionally, we study moduli spaces of flat SL₂(C) connections on Seifert manifolds and their relation to spectra of surface singularities, extending a result of Boden and Curtis for Brieskorn spheres to Seifert rational homology spheres with 3 singular fibers and to Seifert homology spheres with any number of fibers.