Learning Topological Features of $\widehat Z$-invariants

Abstract

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.