$\Delta$ Invariants of Plumbed Manifolds

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.