Untangle knots with motion planning, building off the approach of Ladd & Kavraki (2004). Draw or load a knot diagram and watch it get untangled.
Source code: github.com/zkingston/unknot (web/, MIT).
Hard unknots: Burton et al. (2024), after the original sources listed below; DeepMind’s from Applebaum et al. (2025). Known knots: The Knot Atlas (n.d.). Drawings: SnapPy. Type: Iosevka, SIL Open Font License 1.1. Algorithms: CREDITS.md.
| Space | play / pause (starts a run if none) |
| ← → | step one record; with Shift, 10% of the run |
| V | flip 2D / 3D |
| D / O | draw / open |
| B | best view |
| S | curve / ball and stick |
| R | reset the run |
| Esc | close panel, leave draw mode |
| 3D | drag rotates, wheel zooms, Shift-drag pans, arrows rotate 5°, double-click best view |
Applebaum, T., Blackwell, S., Davies, A., Edlich, T., Juhász, A., Lackenby, M., Tomašev, N., & Zheng, D. (2025). The unknotting number, hard unknot diagrams, and reinforcement learning. Experimental Mathematics. Advance online publication. Data: very hard unknots (CC BY 4.0). doi.org/10.1080/10586458.2025.2542174
Burton, B. A., Chang, H.-C., Löffler, M., Maria, C., de Mesmay, A., Schleimer, S., Sedgwick, E., & Spreer, J. (2024). Hard diagrams of the unknot. Experimental Mathematics, 33(3), 482–500. doi.org/10.1080/10586458.2022.2161676
Curtis, F. (2001). Unknot equivalence. f2.org/maths/kt/unknoteq.html
Freedman, M. H., He, Z.-X., & Wang, Z. (1994). Möbius energy of knots and unknots. Annals of Mathematics, 139(1), 1–50. doi.org/10.2307/2946626
Goeritz, L. (1934). Bemerkungen zur Knotentheorie. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 10(1), 201–210. doi.org/10.1007/BF02940674
Henrich, A., & Kauffman, L. H. (2014). Unknotting unknots. The American Mathematical Monthly, 121(5), 379–390. doi.org/10.4169/amer.math.monthly.121.05.379
Kauffman, L. H., & Lambropoulou, S. (2011). Hard unknots and collapsing tangles. In L. H. Kauffman, S. Lambropoulou, S. Jablan, & J. H. Przytycki (Eds.), Introductory lectures on knot theory (pp. 187–247). World Scientific. doi.org/10.1142/9789814313001_0009
Ladd, A. M., & Kavraki, L. E. (2004). Using motion planning for knot untangling. The International Journal of Robotics Research, 23(7–8), 797–808. doi.org/10.1177/0278364904045469
Ochiai, M. (1990). Non-trivial projections of the trivial knot. Astérisque, 192, 7–10. www.numdam.org/item/AST_1990__192__7_0/
Petronio, C., & Zanellati, A. (2016). Algorithmic simplification of knot diagrams: New moves and experiments. Journal of Knot Theory and Its Ramifications, 25(10), Article 1650059. doi.org/10.1142/S0218216516500590
The Knot Atlas. (n.d.). katlas.org
Tuzun, R. E., & Sikora, A. S. (2018). Verification of the Jones unknot conjecture up to 22 crossings. Journal of Knot Theory and Its Ramifications, 27(3), Article 1840009. doi.org/10.1142/S0218216518400096