  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 $\hat{Z}_b$.
