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SIGMA 19 (2023), 011, 27 pages arXiv:2205.08197
https://doi.org/10.3842/SIGMA.2023.011
Refined and Generalized $\hat{Z}$ Invariants for Plumbed 3-Manifolds
Song Jin Ri ab
a) ICTP, Strada Costiera 11, Trieste 34151, Italy
b) SISSA, Via Bonomea 265, Trieste 34136, Italy
Received September 05, 2022, in final form February 28, 2023; Published online March 19, 2023
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.
Key words: $q$-series; $\hat{Z}$ invariants; plumbed 3-manifolds.
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