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August 26, 2025Open Access

Z and Splice Diagrams

Key Points

The sum of quantum q-series invariants depends only on the splice diagram, showing a unique relationship among manifolds.
Seifert manifolds reveal simple formulas for quantum q-series invariants, enhancing understanding of their properties.
Research employs methods from surface singularities, linking geometry and algebra via splice diagrams and connections.
Implications include a deeper insight into the vanishing of quantum q-series and flat connections on Seifert manifolds.

Abstract

We study quantum q-series invariants of 3-manifolds Z_ 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_ 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_ invariants of Seifert manifolds. Applications include a better understanding of the vanishing of the q-series Z_. 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.

Z and Splice Diagrams

Key Points

Abstract

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