发表机构
Institute of Physics, University of Amsterdam; School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal; Northeastern University; Richard N. Merkin Center for Pure and Applied Mathematics, California Institute of Technology; Ulsan National Institute of Science and Technology; Institute for Mathematics, Academia Sinica; Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学物理研究所; 夸祖鲁-纳塔尔大学数学、统计与计算机科学学院; 东北大学; 加州理工学院理查德·N·默金纯数学与应用数学中心; 蔚山国家科学技术研究院; 中央研究院数学研究所; 阿姆斯特丹大学科特韦格-德弗里斯数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究构建配边3-流形的$\u005cwidehat{Z}$-不变量数据集，用神经网络从$q$-级数系数提取拓扑信息，结合可解释性分析发现其利用谱与几何代理，并揭示该不变量指数与Heegaard Floer $d$-不变量的高精度预测关系，为量子不变量研究提供新方向。
AI 中文摘要
机器学习与数据分析技术近来已成为数学研究中识别模式、提出猜想的有力工具，在低维拓扑领域表现尤为突出。本文提出一种系统性方法，用于处理结构为（截断的）无穷$q$-级数，或等价地说整数无穷级数的数学数据。为应用这一数据分析流程，我们构建了一个关于配边3-流形的$\u005cwidehat{Z}$-不变量（同调块）的综合数据集。我们证明，神经网络能够直接从$q$-级数系数中可靠提取同调类、底层图结构等核心拓扑信息。本研究方法的核心特点是聚焦可解释性；通过对比局部梯度敏感度与全局特征相关性，我们发现网络学会了绕开复杂拓扑规则，转而利用特定的谱代理与几何代理。最后，我们将该流程用于探究同调配边，发现$\u005cwidehat{Z}$-不变量指数与Heegaard Floer $d$-不变量（修正项）之间存在高精度预测关系。这些结果表明，$\u005cwidehat{Z}$-不变量捕捉到了与配边等价相关的精细几何信息，为量子不变量研究开辟了新方向。
英文摘要
Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology. In this paper, we initiate a systematic approach to handling mathematical data structured as (truncated) infinite $q$-series, or equivalently, infinite series of integers. To apply this data analysis pipeline, we construct a comprehensive dataset of $\widehat{Z}$-invariants (homological blocks) for plumbed 3-manifolds. We demonstrate that neural networks can reliably extract essential topological information, such as homology class and underlying graph structure, directly from the $q$-series coefficients. A central feature of our methodology is a focus on interpretability; by contrasting local gradient sensitivity with global feature relevance, we reveal that the networks learn to bypass complex topological rules in favor of specific spectral and geometric proxies. Finally, we apply this pipeline to probe homology cobordism, discovering a high-accuracy predictive relationship between the $\widehat{Z}$-invariant exponents and the Heegaard Floer $d$-invariant (correction term). These results suggest that $\widehat{Z}$-invariants capture subtle geometric information regarding cobordism equivalences, warranting a new direction for the study of quantum invariants.
Comments77 pages, 25 figures
