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To unify connectivity and 3D spatial embedding across heterogeneous scientific data, the authors developed KnottedGraph, a framework that converts complex branched structures into planar diagrams to evaluate topological invariants like the Yamada polynomial. By caching shared partial resolutions and optimizing resolution order, the algorithm scales exact polynomial computation to entangled graphs with up to 500 crossings. Using this high-throughput pipeline to generate invariant datasets, they demonstrate an LLM-assisted discovery loop that successfully derived closed-form analytical Yamada polynomials for both Abelian and non-Abelian graph motif families.
Pushing exact knotted-graph polynomial evaluations to 500 crossings unlocks an automated AI4Math pipeline capable of deriving closed-form topological formulas for complex, entangled graph families.
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.