Refined and Generalized hat Z Invariants for Plumbed 3-Manifolds

IF 0.9 3区 物理与天体物理 Q2 MATHEMATICS
Song Jin Ri
{"title":"Refined and Generalized hat Z Invariants for Plumbed 3-Manifolds","authors":"Song Jin Ri","doi":"10.3842/SIGMA.2023.011","DOIUrl":null,"url":null,"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Symmetry Integrability and Geometry-Methods and Applications","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.3842/SIGMA.2023.011","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1

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.
管道3流形的改进与推广的Z不变量
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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
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.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信