$\mathrm{\Delta }$ Invariants of Plumbed Manifolds

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