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IITM Journal of Information Technology

ISSN (P) 2395-5457 | Single Blind Peer Reviewed Journal

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INSTITUTE OF INNOVATION IN TECHNOLOGY & MANAGEMENT
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Untangling the Complexity: Comparative Analysis of AI/ML Methodologies in Knot Theory

Mariya Ajaz, Priyanka Bhutani
University School of Information, Communication and Technology, GGSIPU, New Delhi

Abstract: The computational challenge of determining whether a tangled loop is topologically equivalent to the unknot is a fundamental benchmark in computational topology. While classical algorithms established this problem in the complexity class NP, the exponential growth of structural data has catalyzed a paradigm shift toward Artificial Intelligence (AI) and Machine Learning (ML) approaches. This paper presents a comparative analysis of contemporary AI/ML techniques applied to the unknotting problem and knot theory at large. We survey foundational and modern studies, mapping the evolution from classical polynomial invariants to modern Deep Reinforcement Learning (DRL) agents and supervised learning models. Furthermore, we conduct a comparative analysis evaluating the inductive biases of different data representations, specifically contrasting sequence-based models with planar graph representations processed via Graph Neural Networks (GNNs). Our analysis extends to real-world applications in molecular biology and robotics, concluding that while sequence models offer computational efficiency, graph-based topological modeling provides superior geometric generalization.

Keywords: Knot Theory, Deep Reinforcement Learning, Graph Neural Networks, Computational Topology, Unknotting

References:

  1. Hass, J., Lagarias, J. C., & Pippenger, N. (1999). The computational complexity of knot and link problems. Journal of the ACM, 46(2), 185-211. https://doi.org/10.1145/301970.301971.
  2. Reidemeister, K. (1927). Elementare Begründung der Knotentheorie. Abhandlungen aus Mathematischen Seminar der Universität Hamburg, 5(1), 24-32. https://doi.org/10.1007/BF02952507.
  3. Li, Z., Ding, K., & Ma, G. (2023). Eigen value knots and their isotopic equivalence in three-state non-Hermitian systems, Physical Review Research, 5(2), 023038, https://doi.org/10.1103/PhysRevResearch.5.023038
  4. Alexander, J. W. (1928). Topological invariants of knots and links. Transactions of the American Mathematical Society, 30(2), 275-306. https://doi.org/10.1090/S0002-9947-1928-1501429-1.
  5. Jones, V. F. R. (1985). A polynomial invariant for knots via von Neumann algebra. Bulletin of the American Mathematical Society (N.S.), 12(1), 103-111. https://doi.org/10.1090/S0273-0979-198515304-2.
  6. Lisitsa, A. (2022). An application of neural networks to a problem in knot theory and group theory (untangling braids) [Preprint]. arXiv. https://arxiv.org/abs/2206.05373.
  7. Gukov, S., Halverson, J., Ruehle, F., & Suwara, P. (2021). Learning to unknot. Machine Learning: Science and Technology, 2(2), Article 025035. https://doi.org/10.1088/2632-2153/abe91f.
  8. Haken, W. (1961). Theorie der Normalflächen: Ein Isotopiekriterium für die Wortprobleme der Gruppentheorie. Acta Mathematica, 105, 245-375. https://doi.org/10.1007/BF02559591.
  9. Vandans, O., Yang, K., Wu, Z., & Dai, L. (2020). Recognizing knots in a polymer conformation by machine learning. Physical Review E, 101(2), Article 022502. https://doi.org/10.1103/PhysRevE.101.022502.
  10. Craven, J. (2023). Learning knot invariants across dimensions. SciPost Physics, 14, Article 021. https://doi.org/10.21468/SciPostPhys.14.2.021.
  11. Craven, J., Hughes, M., Jejjala, V., & Kar, A. (2024). Illuminating new and known relations between knot invariants. Machine Learning: Science and Technology, 5(4), 1-25. https://doi.org/10.1088/26322153/ad95d9.
  12. Davies, A., Veličković, P., Buesing, L., Blackwell, S., Zheng, D., Tomašev, N., Tanburn, R., Battaglia, P., Blundell, C., Juhász, A., Lackenby, M., Geordie, W., 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.
  13. Applebaum, T., Juhász, A., Ruehle, F., & Gukov, S. (2025). The unknotting number, hard unknot diagrams, and reinforcement learning. Experimental Mathematics, 34(1), 1-25. https://doi.org/10.1080/10586458.2025.2542174.
  14. Flapan, E., Henrich, A., & Wong, H. (2022). AlphaFold predicts the most complex protein knot and composite protein knots. Protein Science, 31(8), Article e4380. https://doi.org/10.1002/pro.4380.
  15. Jumper, J., Evans, R., Pritzel, A., Green, T., Figurnov, M., Ronneberger, O., Tunyasuvunakool, K., Bates, R., Zídek, A., Potapenko, A., Bridgland, A., Meyer, C., Kohl, S. A. A., Ballard, A. J., Cowie, A., RomeraParedes, B., Stanway, S., Faneva, R., Vinyals, O., ... Hassabis, D. (2021). Highly accurate protein structure prediction with AlphaFold. Nature, 596(7873), 583-589. https://doi.org/10.1038/s41586-02103819-2.
  16. Hang, K., Lerrel, P., Wang, S., Abbeel, P., & Solowjow, E. (2023). SGTM 2.0: Autonomously untangling long cables using interactive perception. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA) (pp. 1-7). IEEE. https://doi.org/10.1109/ICRA48891.2023.10161234.
  17. Driggs, N. (2025). Exploring representations and inductive bias for machine learning tasks in knot theory [Master's thesis, Brigham Young University]. BYU Scholars Archive. https://scholarsarchive.byu.edu/etd/10813/.
  18. Lindsay, A. (2025). On the learnability of knot invariants: Representation, predictability, and neural similarity [Preprint]. arXiv. https://arxiv.org/abs/2502.12243.
  19. Ruehle, F. (2025). Learning topological invariance [Preprint]. arXiv. https://arxiv.org/abs/2504.12390.
  20. Zhang, Z., Zhu, Y., & Dai, L. (2025). Recognizing and generating knotted molecular structures by machine learning [Preprint]. arXiv. https://arxiv.org/abs/2501.12780.
  21. de Boer, J., Sazdanovic, R., & Suwara, P. (2024). A phenomenological approach to interactive knot diagrams. In Proceedings of IEEE VIS (pp. 1-5). IEEE. https://doi.org/10.1109/VIS54894.2024.00010.
  22. Al-Amayreh, M., & Salles, M. (2022). Untangling braids with deep reinforcement learning. In Proceedings of the 35th International Florida Artificial Intelligence Research Society Conference (FLAIRS). https://doi.org/10.32473/flairs.v35i1.130657.
  23. Quantinuum. (2024, May 22). Untangling the mysteries of knots with quantum computers. Quantinuum Blog. https://www.quantinuum.com/blog/untangling-the-mysteries-of-knots-with-quantum-computers.
  24. Joshi, R., Sastry, K., Bharathi, M., & Bhutani, P. (2023). Smart cities integrating artificial intelligence and IoT: A review. International Journal of Computer Applications, 182(47), 1-5.
  25. Jejjala, V., Kar, A., & Parrikar, O. (2019). Deep learning the hyperbolic volume of a knot. Physics Letters B, 799, Article 135033. https://doi.org/10.1016/j.physletb.2019.135033.
  26. Jaretzki, L. (2023). Geometric deep learning approach to knot theory. arXiv. https://doi.org/10.48550/arXiv.2305.16808.
  27. Laakkonen, T., Rinaldi, E., Self, C. N., Chertkov, E., DeCross, M., Hayes, D., ... & Meichanetzidis, K. (2025). Less quantum, more advantage: An end-to-end quantum algorithm for the Jones polynomial. arXiv. https://doi.org/10.48550/arXiv.2503.05625.
  28. Schmidhuber, A., Reilly, M., Zanardi, P., Lloyd, S., & Lauda, A. (2025). A quantum algorithm for Khovanov homology. arXiv. https://doi.org/10.48550/arXiv.2501.12378.
  29. Jackson, N. (2013). Evolution of unknotting strategies for knots and braids [Preprint]. arXiv. https://arxiv.org/abs/1302.0787.
  30. Lackenby, M. (2020). Unknot recognition in quasi-polynomial time. Foundations of Computational Mathematics, 20(5), 1141-1179. https://doi.org/10.1007/s10208-020-09455-8.
  31. Verma, N., Bhutani, P., Lalit, R., & Venugopal, S. (2025). Map reduce framework-assisted feature analysis and adaptive multiplicative Bi-RNN using big data analytics for decision-making. International Journal of Computational Intelligence Systems, 18(1), 1-30.
  32. Gukov, S. (2023, May). Machine learning and hard problems in topology [Workshop presentation]. ICERM Topical Workshop on the Mathematics of Knots, Brown University, Providence, RI, United States. https://icerm.brown.edu/program/topical_workshop/tw-23-tkt.
  33. Dhaka, P., Sehrawat, R., & Bhutani, P. (2023). An innovative approach to cardiovascular disease prediction: A hybrid deep learning model. Engineering, Technology & Applied Science Research, 13(6), 12396-12403. https://doi.org/10.48084/etasr.6503.
  34. Sazdanovic, R. (2025). Data driven perspectives on knot theory [Preprint]. arXiv. https://arxiv.org/abs/2503.15103.
  35. Gukov, S., Halverson, J., Manolescu, C., & Ruehle, F. (2025). Searching for ribbons with machine learning. Machine Learning: Science and Technology, 6(2), 1-25. https://doi.org/10.1088/26322153/ade362.
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