Knottedgraph enables scalable topological analysis of complex scientific networks
KnottedGraph: Scalable knotted-graph topology for scientific and mathematical discovery
Mathematical SoftwareComputational GeometrySymbolic Computation
Summary
Scientific data often involves complex shapes and connections that are hard to study all at once. The authors created KnottedGraph, a tool that turns these complex data structures into knotted graphs that keep both how parts connect and their 3D layout. This makes it possible to perform advanced mathematical analyses to classify and understand these structures more easily. Their method is efficient enough to handle very complex graphs and helps discover new mathematical formulas with the help of AI.
What this means in practice
- •For data scientists in physical sciences: Classify and analyze complex spatial network data by converting it to knotted graphs with preserved topology and embedding.
- •For mathematical software developers: Build tools that compute topological invariants efficiently for large knotted graphs to support mathematical and scientific research.
Authors
Hakan Akgün, Xianquan Yan, Kehan Liu, Zhaoyun Chen, Ching Hua Lee
Abstract
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.