管道3流形的改进与推广的Z不变量

IF 0.9 3区 物理与天体物理 Q2 MATHEMATICS Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-05-17 DOI:10.3842/SIGMA.2023.011
Song Jin Ri
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引用次数: 1

摘要

我们引入了一个双变量精化$\hat{Z}_a铅垂3-流形不变量$\hat的(q,t)${Z}_a(q) $,其先前被定义为弱负定铅垂3-流形。我们还提供了一些明确的例子,其中我们论证了获得$\hat的恢复过程{Z}_a(q) $\hat中的${Z}_a(q,t)$。对于具有两个高价顶点的铅垂3-流形,我们利用二元二次丢番图方程的显式整数解解析计算了极限。基于回收$\hat的数值计算{Z}_a(q) 对于具有两个高价顶点的铅垂,我们提出了一个猜想,即恢复的$\hat{Z}_a(q) $,如果存在的话,是所有树铅垂3流形的不变量。最后,我们提供了$\hat的公式{Z}_a(q,t)$,以组件的形式表示的3个管道歧管的连接总和。
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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.
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来源期刊
CiteScore
1.80
自引率
0.00%
发文量
87
审稿时长
4-8 weeks
期刊介绍: Scope Geometrical methods in mathematical physics Lie theory and differential equations Classical and quantum integrable systems Algebraic methods in dynamical systems and chaos Exactly and quasi-exactly solvable models Lie groups and algebras, representation theory Orthogonal polynomials and special functions Integrable probability and stochastic processes Quantum algebras, quantum groups and their representations Symplectic, Poisson and noncommutative geometry Algebraic geometry and its applications Quantum field theories and string/gauge theories Statistical physics and condensed matter physics Quantum gravity and cosmology.
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