论文标题
精制和广义的$ \ hat {z} $不变的3型物体
Refined and Generalized $\hat{Z}$ Invariants for Plumbed 3-Manifolds
论文作者
论文摘要
我们介绍了一个两变量的完善$ \ hat {z} _a（q，t）$的3个manifold不变式$ \ hat {z} _a（q）$，以前是针对弱负的确定液化的3个manifolds定义的。我们还提供了许多明确的示例，在其中，我们认为从$ \ hat {z} _a _a（q，t）$获得$ \ hat {z} _a（q）$获得$ \ hat {z} _a（q）$，通过限制$ t \ rightarrow 1 $。对于具有两个高价值顶点的3个脉冲的3个manifolds，我们通过在两个变量中使用二次二磷甘氨酸方程的显式整数解决方案来分析限制。基于带有两个高价值顶点的管道的回收的$ \ hat {z} _a（q）$的数值计算，我们提出一个猜想，即恢复的$ \ hat {z} _a（q）$，如果存在，则是所有树木液化3-manifolds的不变。最后，我们提供了一个$ \ hat {z} _a（q，t）$的公式，以根据组件的连接3个manifolds的连接总和。
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.
