SIGMA 21 (2025), 091, 21 pages      arXiv:2412.02042      https://doi.org/10.3842/SIGMA.2025.091

$\Delta$ Invariants of Plumbed Manifolds

Shimal Harichurn ^a, András Némethi ^bcde and Josef Svoboda ^f
a) School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, South Africa
^b) Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, 1053 Budapest, Hungary
^c) Department of Mathematics, University of Budapest (ELTE), Pázmány Péter Sétány 1/A, 1117, Budapest, Hungary
^d) Babeş-Bolyai University, Str. M. Kogălniceanu 1, 400084 Cluj-Napoca, Romania
^e) Basque Center for Applied Mathematics (BCAM), Alameda de Mazarredo 14, 48009 Bilbao, Spain
^f) Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA

Received February 15, 2025, in final form October 17, 2025; Published online October 24, 2025

Abstract
We study the minimal $q$-exponent $\Delta$ in the BPS $q$-series $\widehat{Z}$ of negative definite plumbed 3-manifolds equipped with a spin$^{\rm c}$-structure. We express $\Delta$ of Seifert manifolds in terms of an invariant commonly used in singularity theory. We provide several examples illustrating the interesting behaviour of $\Delta$ for non-Seifert manifolds. Finally, we compare $\Delta$ invariants with correction terms in Heegaard-Floer homology.

Key words: 3-manifold topology; quantum invariant; $q$-series; splice diagram.

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