$\hat{Z}$ and Splice Diagrams

Abstract

We study quantum $q$-series invariants of 3-manifolds ${\hat{Z}}_{\ddot{I}}ƒ$ 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 ${\hat{Z}}_{\ddot{I}}ƒ$ 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 ${\hat{Z}}_{\ddot{I}}ƒ$ invariants of Seifert manifolds. Applications include a better understanding of the vanishing of the $q$-series ${\hat{Z}}_{\ddot{I}}ƒ$. Additionally, we study moduli spaces of flat ${\mathrm{SL}}_2(\mathbb{C})$ 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.