KnottedGraph: Scalable knotted-graph topology for scientific and mathematical discovery
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.
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BibTeX
@misc{akgun2026knottedgraph,
title = {{KnottedGraph: Scalable knotted-graph topology for scientific and mathematical discovery}},
author = {Hakan Akgün and Xianquan Yan and Kehan Liu and Zhaoyun Chen and Ching Hua Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31152},
eprint = {2609.31152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31152},
}
RIS
TY - UNPB
TI - KnottedGraph: Scalable knotted-graph topology for scientific and mathematical discovery
AU - Hakan Akgün
AU - Xianquan Yan
AU - Kehan Liu
AU - Zhaoyun Chen
AU - Ching Hua Lee
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31152
UR - https://arxiv.org/abs/2609.31152
AB - 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.
ER -