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

TL;DR

This paper introduces a two-variable refinement of plumbed 3-manifold invariants, explores their recovery process, and conjectures their invariance across all tree plumbed 3-manifolds, with explicit examples and formulas.

Contribution

It defines a new two-variable invariant $ ilde{Z}_a(q,t)$ for plumbed 3-manifolds, analyzes its limit to recover existing invariants, and proposes conjectures on their invariance.

Findings

Explicit formulas for $ ilde{Z}_a(q,t)$ for certain plumbed 3-manifolds.

Numerical evidence supporting invariance conjecture for all tree plumbed 3-manifolds.

A formula for $ ilde{Z}_a(q,t)$ under connected sum operations.

Abstract

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.