Refined and Generalized $\hat{Z}$ Invariants for Plumbed $3$-Manifolds

We introduce a two-variable refinement $\hat{Z}_a(q,t)$ of plumbed $3$-manifold invariants $\hat{Z}_a(q)$, which were previously defined for weakly negative definite plumbed $3$-manifolds. We also provide a number of explicit examples in which we argue the recovering process to obtain $\hat{Z}_a(q)$ from $\hat{Z}_a(q,t)$ by taking a limit $ t\rightarrow 1 $. For plumbed $3$-manifolds with two high-valency vertices, we analytically compute the limit by using the explicit integer solutions of quadratic Diophantine equations in two variables. Based on numerical computations of the recovered $\hat{Z}_a(q)$ for plumbings with two high-valency vertices, we propose a conjecture that the recovered $\hat{Z}_a(q)$, if exists, is an invariant for all tree plumbed $3$-manifolds. Finally, we provide a formula of the $\hat{Z}_a(q,t)$ for the connected sum of plumbed $3$-manifolds in terms of those for the components.