$\Delta$ Invariants of Plumbed Manifolds

Shimal Harichurn^a, András Némethi^bcde, Josef Svoboda<sup>f

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

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$^{\mathrm c}$-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.

Keywords: $3$-manifold topology, quantum invariant, $q$-series, splice diagram.

MSC: 57K30, 57K31, 57K16

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

Language: English

DOI: 10.3842/SIGMA.2025.091

ArXiv: 2412.02042