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README.md
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@@ -70,4 +70,36 @@ Every catalogued knot's embedding, flattened with UMAP and coloured by signature
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- Prime knots up to 13 crossings only, and only the chirality KnotInfo lists; a mirror image is a different input.
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- Nearest-neighbour naming is a guess, not a proof. Use SnapPy or KnotInfo to confirm.
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- Diagrams over 64 crossings are out of range.
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- Prime knots up to 13 crossings only, and only the chirality KnotInfo lists; a mirror image is a different input.
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- Nearest-neighbour naming is a guess, not a proof. Use SnapPy or KnotInfo to confirm.
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- Diagrams over 64 crossings are out of range.
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## 📚 Related work
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The closest work is Halverson & Ruehle (2025), who train contrastive and generative models to embed braid words of the same knot at the same point. Knot2Vec works on PD codes scrambled by Reidemeister moves instead, covers every prime knot up to 13 crossings, and holds out 10% of the knots.
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### Articles
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- Applebaum, T., Blackwell, S., Davies, A., Edlich, T., Juhász, A., Lackenby, M., Tomašev, N., & Zheng, D. (2024). *The unknotting number, hard unknot diagrams, and reinforcement learning* (arXiv:2409.09032). arXiv. https://doi.org/10.48550/arXiv.2409.09032
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- Craven, J., Hughes, M., Jejjala, V., & Kar, A. (2022). *(K)not machine learning* (arXiv:2201.08846). arXiv. https://doi.org/10.48550/arXiv.2201.08846
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- Davies, A., Juhász, A., Lackenby, M., & Tomašev, N. (2024). The signature and cusp geometry of hyperbolic knots. *Geometry & Topology, 28*(5), 2313–2343. https://doi.org/10.2140/gt.2024.28.2313
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- Davies, A., Veličković, P., Buesing, L., Blackwell, S., Zheng, D., Tomašev, N., Tanburn, R., Battaglia, P., Blundell, C., Juhász, A., Lackenby, M., Williamson, G., Hassabis, D., & Kohli, P. (2021). Advancing mathematics by guiding human intuition with AI. *Nature, 600*(7887), 70–74. https://doi.org/10.1038/s41586-021-04086-x
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- Gukov, S., Halverson, J., Ruehle, F., & Sułkowski, P. (2021). Learning to unknot. *Machine Learning: Science and Technology, 2*(2), 025035. https://doi.org/10.1088/2632-2153/abe91f
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- Halverson, J., & Ruehle, F. (2025). *Learning topological invariance* (arXiv:2504.12390). arXiv. https://doi.org/10.48550/arXiv.2504.12390
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- Hughes, M. C. (2020). A neural network approach to predicting and computing knot invariants. *Journal of Knot Theory and Its Ramifications, 29*(3), 2050005. https://doi.org/10.1142/S0218216520500054
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- Jaretzki, L. (2023). *Geometric deep learning approach to knot theory* (arXiv:2305.16808). arXiv. https://doi.org/10.48550/arXiv.2305.16808
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- Jejjala, V., Kar, A., & Parrikar, O. (2019). Deep learning the hyperbolic volume of a knot. *Physics Letters B, 799*, 135033. https://doi.org/10.1016/j.physletb.2019.135033
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- Mihajlovic, D., & Michieletto, D. (2026). *Shortcut learning in geometric knot classification* (arXiv:2602.17350). arXiv. https://doi.org/10.48550/arXiv.2602.17350
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- Piccirillo, L. (2020). The Conway knot is not slice. *Annals of Mathematics, 191*(2), 581–591. https://doi.org/10.4007/annals.2020.191.2.5
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- Sleiman, J. L., Conforto, F., Fosado, Y. A. G., & Michieletto, D. (2024). Geometric learning of knot topology. *Soft Matter, 20*(1), 71–78. https://doi.org/10.1039/D3SM01199B
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### Data and software
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- [KnotInfo](https://knotinfo.org/): table of knot invariants by C. Livingston and A. H. Moore, and its Python package [database_knotinfo](https://pypi.org/project/database-knotinfo/).
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- [SnapPy](https://snappy.computop.org/): M. Culler, N. M. Dunfield, M. Goerner and J. R. Weeks' program for the topology and geometry of 3-manifolds, with the spherogram link library.
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### Blogs and press
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- DeepMind (2021). [Exploring the beauty of pure mathematics in novel ways](https://deepmind.google/discover/blog/exploring-the-beauty-of-pure-mathematics-in-novel-ways/).
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- IAIFI (2025). [Learning topological invariance](https://research.iaifi.org/posts/learning-topological-invariance).
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- IAIFI. [Learning to unknot](https://research.iaifi.org/posts/learning-to-unknot).
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- Klarreich, E. (2020). [Graduate student solves decades-old Conway knot problem](https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/). Quanta Magazine.
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- Quanta Magazine (2022). [Why mathematicians study knots](https://www.quantamagazine.org/why-mathematicians-study-knots-20221031/).
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