Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 073, 30 pages      arXiv:2304.00699      https://doi.org/10.3842/SIGMA.2025.073

$\widehat{Z}$ and Splice Diagrams

Sergei Gukov a, Ludmil Katzarkov b and Josef Svoboda a
a) Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA
b) Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA

Received November 22, 2024, in final form August 16, 2025; Published online August 26, 2025

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

Key words: 3-manifold topology; quantum invariant; surface singularity; splice diagram.

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