Advertisement

On a flow to untangle elastic knots

  • Chun-Chi Lin
  • Hartmut R. Schwetlick
Article

Abstract

In this paper we present a method to untangle smooth knots by a gradient flow for a suitable energy. We show that the flow of smooth initial knots remains smooth for all time and approaches asymptotically an “optimal embedding” in its isotopy type. The method is to set up a gradient flow for the total energy of knots, which consists of bending energy and the Möbius energy of knots.

Mathematics Subject Classification (2000)

35K55 57M25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abrams A., Cantarella J., Fu J.H.G., Ghomi M., Howard R.: Circles minimize most knot energies. Topology 42(2), 381–394 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aubin T.: Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften, 252. Springer-Verlag, New York (1982)Google Scholar
  3. 3.
    Buck G., Simon J.: Thickness and crossing number of knots. Topol. Appl. 91(3), 245–257 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    do Carmo M.P.: Differential geometry of curves and surfaces, translated from the Portuguese. Prentice-Hall, Inc., Englewood Cliffs, NJ (1976)Google Scholar
  5. 5.
    Dziuk G., Kuwert E., Schätzle R.: Evolution of elastic curves in \({\mathbb{R}^{n}}\) , existence and computation. SIAM J. Math. Anal. 33(5), 1228–1245 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Eidel’man S.D.: Parabolic equations. In: Egerov, Yu.V., Shubinv, M.A. (eds) Encyclopaedia of mathematical sciences, Scientific Publishers, Berlin, Heidelberg, New York (1994)Google Scholar
  7. 7.
    Freedman M.H., He Z.X., Wang Z.: Möbius energy of knots and unknots. Ann. Math. (2) 139(1), 1–50 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gonzalez O., Maddocks J.H.: Global curvature, thickness, and the ideal shapes of knots. Proc. Natl. Acad. Sci. USA 96(9), 4769–4773 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7(1), 65–222 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hatcher A.E.: A proof of a Smale conjecture, \({{\rm Diff}(S^{3})\simeq {\rm O}(4)}\) . Ann. Math. (2) 117(3), 553–607 (1983)CrossRefMathSciNetGoogle Scholar
  11. 11.
    He Z.X.: The Euler–Lagrange equation and heat flow for the Möbius energy. Comm. Pure Appl. Math. 53(4), 399–431 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Huang, M., Grzeszczuk, R.P., Kauffman, L.H.: Untangling knots by stochastic energy optimization. In: Proceedings of the 7th IEEE visualization conference, IEEE (1996)Google Scholar
  13. 13.
    Kusner, R.B., Sullivan, J.M.: Möbius-invariant knot energies, Ideal knots, 315–352, Ser. Knots Everything, 19, World Sci. Publ., River Edge, NJ, (1998)Google Scholar
  14. 14.
    Langer J., Singer D.A.: Knotted elastic curves in \({\mathbb{R}^3}\) . J. Lond. Math. Soc. (2) 30(3), 512–520 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Langer J., Singer D.A.: The total squared curvature of closed curves. J. Differ. Geom. 20(1), 1–22 (1984)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Langer J., Singer D.A.: Curve straightening and a minimax argument for closed elastic curves. Topology 24(1), 75–88 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ligocki T.J., Sethian J.A.: Recognizing knots using simulated annealing. J. Knot Theory Ramif. 3(4), 477–495 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lin C.C., Schwetlick H.R.: On the geometric flow of Kirchhoff elastic rods. SIAM J. Appl. Math. 65(2), 720–736 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    O’Hara J.: Energy of a knot. Topology 30(2), 241–247 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    O’Hara J.: Energy functionals of knots. II. Topol. Appl. 56(1), 45–61 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    O’Hara J.: Energy of knots and conformal geometry, Series on Knots and Everything, 33. World Scientific Publishing Co., Inc., River Edge, NJ (2003)CrossRefGoogle Scholar
  22. 22.
    Polden, A.: Curves and surfaces of least total curvature and fourth-order flows. Ph.D. dissertation, Universität Tübingen, Tübingen, Germany (1996)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsNational Taiwan Normal UniversityTaipeiTaiwan
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK

Personalised recommendations