$\hat{Z}_b$ for plumbed manifolds and splice diagrams

Abstract: We study $q$-series invariants of 3-manifolds $\hat{Z}_b$ defined by Gukov--Pei--Putrov--Vafa using techniques from the theory of normal surface singularities such as splice diagrams. This provides a link between algebraic geometry with quantum topology. We show that the (suitably normalized) sum of all $\hat{Z}_b$ depends only on the splice diagram, and in particular, it agrees for manifolds with the same universal Abelian cover. Using these ideas we find simple formulas for $\hat{Z}_b$ invariants of Seifert manifolds that resemble equivariant Poincar\'e series of corresponding quasihomogeneous singularity. Applications include a better understanding of the vanishing of the $q$-series $\hat{Z}_b$.