KnottedGraph：面向科学与数学发现的可扩展纽结图拓扑
KnottedGraph: Scalable knotted-graph topology for scientific and mathematical discovery
- National University of Singapore（新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出KnottedGraph框架，将科学数据转换为纽结图以同时保留连通性和空间嵌入，通过可扩展的Yamada多项式评估支持大规模拓扑分类，并利用LLM辅助从拓扑数据中发现闭式公式，实现AI4Math数学发现。
AI中文摘要：
科学数据涵盖异构结构，包括坐标、网络、曲面、体积和场，然而其拓扑可以通过图连通性、环结构、亏格和空间嵌入在通用框架内进行量化。基于图和同调的摘要无法确定空间嵌入，而标准的纽结和链环多项式需要扩展以适应分支图。在此，我们引入KnottedGraph，一个计算框架，将此类科学表示转换为保留图连通性和空间嵌入的纽结图。它构建投影图和PD码，支持各种拓扑分析，包括用于拓扑分类的Yamada多项式评估。为了实现可扩展的精确评估，它结合了留下相同未解析连接的部分解析，并优化其处理顺序；所得算法针对已发表的、具有多达500个交叉的纽结图拓扑不变量进行了验证。这种可扩展性使我们能够引入一种LLM辅助的数学发现方法，其中利用跨纽结图族生成的计算拓扑数据来识别候选的闭式公式。通过这种方法，我们为表现出阿贝尔和非阿贝尔词序列的通用图基序族确定了解析Yamada多项式。总之，这些可扩展能力使纽结图拓扑在科学领域内计算上可访问，实现大规模分类，并开辟了从拓扑数据到LLM辅助AI4Math发现的途径。
英文摘要：
Scientific data span heterogeneous structures, including coordinates, networks, surfaces, volumes and fields, yet their topology can be quantified within a common framework through graph connectivity, cycle structure, genus and spatial embedding. Graph- and homology-based summaries do not determine spatial embedding, while standard knot and link polynomials require extensions to accommodate branching graphs. Here, we introduce KnottedGraph, a computational framework that converts such scientific representations to knotted graphs that retain graph connectivity and spatial embedding together. It constructs projected diagrams and PD codes, enabling various topological analyses, including Yamada-polynomial evaluation for topological classification. For scalable exact evaluation, it combines partial resolutions that leave the same unresolved connections and optimizes their processing order; the resulting algorithm is verified against published topological invariants of knotted graphs with up to 500 crossings. This scalability enables us to introduce an LLM-assisted mathematical-discovery methodology, in which computational topological data generated across knotted-graph families are used to identify candidate closed-form formulas. With this approach, we identify analytical Yamada-polynomials for generic graph motif families exhibiting Abelian and non-Abelian word sequences. Together, these scalable capabilities make knotted-graph topology computationally accessible across scientific domains, enabling large-scale classification and introducing a route from topological data to LLM-assisted AI4Math discovery.
