Refined and Generalized Z Invariants for Plumbed 3-Manifolds
Abstract
We introduce a two-variable refinement Za(q,t) of plumbed 3-manifold invariants Za(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 Za(q) from Za(q,t) by taking a limit t→ 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 Za(q) for plumbings with two high-valency vertices, we propose a conjecture that the recovered Za(q), if exists, is an invariant for all tree plumbed 3-manifolds. Finally, we provide a formula of the Za(q,t) for the connected sum of plumbed 3-manifolds in terms of those for the components.
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