Instructions: click on blue and green circles until the number of crossings in the picture is minimal possible (see 'Minimal possible length' below).
Both types of circles correspond to conjugation moves $f\mapsto s_{i}f{s}_i$, but blue circles preserve the number $\ell (f)$ of crossings, and green circles decrease it by 2.
Press the "Run ODE" button to solve the problem using a system of linear ODE: $$x_i(0)=i;x_i^{\prime }(t)=x_{f(i)}-x_i\mathrm{\forall }i\in \mathbb{Z},$$ where $x_i(t)$ is the horizontal coordiante of the two points labeled $i$ at the top and bottom row of the picture. See the proof of Proposition 4.13 in arXiv:2212.12962 for more details.

$[f(1),f(2),...,f(n)]=$ Plot undo Run ODE

Length: $\ell (f)=$ 4. Minimal possible length: 0. $k(f)=$ 1, $n(f)=$ 4. Number of cycles: $\mathrm{ncyc}(f)=$ 1. Choose a level: Prev < 1. Easy (n=5, ncyc=1, difficulty=4) 2. Easy (n=5, ncyc=1, difficulty=5) 3. Easy (n=5, ncyc=1, difficulty=6) 4. Easy (n=6, ncyc=2, difficulty=7) 5. Easy (n=6, ncyc=1, difficulty=8) 6. Easy (n=6, ncyc=1, difficulty=9) 7. Easy (n=6, ncyc=1, difficulty=11) 8. Easy (n=6, ncyc=2, difficulty=12) 9. Easy (n=7, ncyc=1, difficulty=12) 10. Easy (n=7, ncyc=1, difficulty=13) 11. Easy (n=7, ncyc=2, difficulty=14) 12. Medium (n=7, ncyc=2, difficulty=15) 13. Medium (n=7, ncyc=2, difficulty=16) 14. Medium (n=7, ncyc=1, difficulty=17) 15. Medium (n=7, ncyc=2, difficulty=18) 16. Medium (n=7, ncyc=1, difficulty=19) 17. Medium (n=8, ncyc=2, difficulty=21) 18. Medium (n=7, ncyc=3, difficulty=26) 19. Medium (n=8, ncyc=2, difficulty=26) 20. Medium (n=9, ncyc=1, difficulty=29) 21. Medium (n=9, ncyc=2, difficulty=29) 22. Medium (n=9, ncyc=1, difficulty=30) 23. Medium (n=9, ncyc=3, difficulty=33) 24. Medium (n=9, ncyc=2, difficulty=34) 25. Medium (n=8, ncyc=2, difficulty=36) 26. Medium (n=9, ncyc=2, difficulty=37) 27. Medium (n=8, ncyc=1, difficulty=38) 28. Medium (n=9, ncyc=1, difficulty=39) 29. Medium (n=9, ncyc=3, difficulty=39) 30. Medium (n=9, ncyc=2, difficulty=40) 31. Medium (n=9, ncyc=3, difficulty=40) 32. Medium (n=9, ncyc=1, difficulty=41) 33. Medium (n=9, ncyc=2, difficulty=42) 34. Medium (n=9, ncyc=3, difficulty=43) 35. Medium (n=8, ncyc=2, difficulty=45) 36. Medium (n=9, ncyc=2, difficulty=45) 37. Medium (n=9, ncyc=1, difficulty=47) 38. Medium (n=9, ncyc=1, difficulty=49) 39. Medium (n=9, ncyc=1, difficulty=58) 40. Hard (n=10, ncyc=2, difficulty=64) 41. Hard (n=9, ncyc=3, difficulty=66) 42. Hard (n=10, ncyc=1, difficulty=67) 43. Hard (n=10, ncyc=1, difficulty=75) 44. Hard (n=11, ncyc=3, difficulty=75) 45. Hard (n=11, ncyc=1, difficulty=76) 46. Hard (n=9, ncyc=2, difficulty=77) 47. Hard (n=11, ncyc=3, difficulty=78) 48. Hard (n=10, ncyc=2, difficulty=87) 49. Hard (n=10, ncyc=4, difficulty=102) 50. Hard (n=10, ncyc=1, difficulty=120) > Next

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