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On rectifiable curves with L p-bounds on global curvature: self-avoidance, regularity, and minimizing knots

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Abstract

We discuss the analytic properties of curves γ whose global curvature function ρ G [γ]−1 is p-integrable. It turns out that the L p-norm \({\mathcal{U}_{p}(\gamma):=\|\rho_G[\gamma]^{-1}\|_{L^p}}\) is an appropriate model for a self-avoidance energy interpolating between “soft” knot energies in form of singular repulsive potentials and “hard” self-obstacles, such as a lower bound on the global radius of curvature introduced by Gonzalez and Maddocks. We show in particular that for all p > 1 finite \({\mathcal{U}_{p}}\) -energy is necessary and sufficient for W 2,p-regularity and embeddedness of the curve. Moreover, compactness and lower-semicontinuity theorems lead to the existence of \({\mathcal{U}_p}\) -minimizing curves in given isotopy classes. There are obvious extensions to other variational problems for curves and nonlinearly elastic rods, where one can introduce a bound on \({\mathcal{U}_{p}}\) to preclude self-intersections.

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References

  1. Abrams A., Cantarella J., Fu J.H.G., Ghomi M. and Howard R. (2003). Circles minimize most knot energies. Topology 42: 381–394

    Article  MATH  MathSciNet  Google Scholar 

  2. Banavar, J.R., Flammini, A., Marenduzzo, D., Maritan, A., Trovato, A.: Geometry of compact tubes and protein structures. ComPlexUs 1 (2003)

  3. Banavar J.R., Gonzalez O., Maddocks J.H. and Maritan A. (2003). Self-interactions of strands and sheets. J. Statist. Phys. 110: 35–50

    Article  MATH  MathSciNet  Google Scholar 

  4. Banavar J.R., Maritan A. and Micheletti, C. Trovato A. (2002). Geometry and physics of proteins. Proteins 47: 315–322

    Article  Google Scholar 

  5. Brylinski J.-L. (1999). The beta function of a knot. Int. Math. 10: 415–423

    Article  MATH  MathSciNet  Google Scholar 

  6. Buck, G., Simon, J.: Energy and length of knots. In: Suzuki (ed.) Lectures at knots 96, pp. 219–234, World Scientific, Singapore (1997)

  7. Cantarella J., Fu J.H.G., Kusner R.B., Sullivan J.M. and Wrinkle N.C. (2006). Criticality for the Gehring link problem. Geometry and Topology 10: 2055–2061

    Article  MATH  MathSciNet  Google Scholar 

  8. Cantarella J., Kusner R.B. and Sullivan J.M. (2002). On the minimum ropelength of knots and links. Inv. math. 150: 257–286

    Article  MATH  MathSciNet  Google Scholar 

  9. Cantarella, J., Piatek, M., Rawdon, E.: Visualizing the tightening of knots. In: VIS’05: Proc. of the 16th IEEE Visualization 2005, pp. 575–582. IEEE Computer Society, Washington (2005)

  10. Carlen, M., Laurie, B., Maddocks, J.H., Smutny, J.: Biarcs, global radius of curvature, and the computation of ideal knot shapes. In: Calvo, Millett, Rawdon, Stasiak (eds.) Physical and Numerical Models in Knot Theory, pp. 75–108. Ser. on Knots and Everything, vol. 36. World Scientific, Singapore (2005)

  11. Cloiseaux, G., Jannink, J.F.: Polymers in Solution: their Modeling and Structure. Clarendon Press, Oxford (1990)

  12. Delrow J.J., Gebe J.A. and Schurr J.M. (1997). Comparison of hard-cylinder and screened Coulomb interaction in the modeling of supercoiled DNAs. Biophysics 42: 455–470

    Google Scholar 

  13. Doi M. and Edwards S.F. (1993). The Theory of Polymer Dynamics. Clarendon Press, New York

    Google Scholar 

  14. Durumeric, O.C.: Local structure of ideal shapes of knots. ar**v:math.DG/0204063 (2002)

  15. Durumeric, O.C.: Thickness formula and C 1-compactness for C 1,1 Riemannian submanifolds. ar**v:math.DG/0204050 (2002)

  16. Freedman M.H., He Z.-X. and Wang Z. (1994). Möbius energy of knots and unknots. Ann. Math. 139(2): 1–50

    MATH  MathSciNet  Google Scholar 

  17. Fukuhara, S.: Energy of a knot. In: A fête of Topology, pp. 443–451. Academic, Boston (1988)

  18. Gerlach, H.: Global curvature for open and closed curves in \({\mathbb{R}^n}\) . Diplom thesis, Univ. Bonn (2004)

  19. Gonzalez O. and de la Llave R. (2003). Existence of ideal knots. J. Knot Theory Ramifications 12: 123–133

    Article  MATH  MathSciNet  Google Scholar 

  20. Gonzalez O. and Maddocks J.H. (1999). Global curvature, thickness and the ideal shape of knots. In: Proc. Nat. Acad. Sci. USA 96(9): 4769–4773

    Article  MATH  MathSciNet  Google Scholar 

  21. Gonzalez O., Maddocks J.H. and Smutny J. (2002). Curves, circles and spheres. Contemp. Math. 304: 195–215

    MathSciNet  Google Scholar 

  22. Gonzalez O., Maddocks J.H., Schuricht F. and von der Mosel H. (2002). Global curvature and self-contact of nonlinearly elastic curves and rods. Calc. Var. Partial Diff. Equ. 14: 29–68

    Article  MATH  MathSciNet  Google Scholar 

  23. Hajłasz P. (1996). Sobolev spaces on an arbitrary metric space. Potential Anal. 6: 413–415

    Google Scholar 

  24. Hajłasz P. (2003). A new characterization of the Sobolev space. Studia Math. 159: 263–275

    Article  MathSciNet  MATH  Google Scholar 

  25. Hahlomaa I. (2005). Menger curvature and Lipschitz parametrizations in metric spaces. Fund. Math. 185: 143–169

    Article  MATH  MathSciNet  Google Scholar 

  26. Hahlomaa, I.: Curvature integral and Lipschitz parametrization in 1-regular metric spaces. Preprint (2005)

  27. He Z.-X. (2000). The Euler-Lagrange equation and heat flow for the Möbius energy. Comm. Pure Appl. Math. 53: 399–431

    Article  MATH  MathSciNet  Google Scholar 

  28. Katritch V., Bednar J., Michoud D., Scharein R.G., Dubochet J. and Stasiak A. (1996). Geometry and physics of knots. Nature 384: 142–145

    Article  MathSciNet  Google Scholar 

  29. Kusner, R. B., Sullivan, J.M.: Möbius-invariant knot energies. In: Stasiak, Katritch, Kauffman (eds.) Ideal knots, pp. 315–352. Ser. on Knots and Everything, vol. 19, World Scientific, River Edge, (1998)

  30. Kusner, R.B., Sullivan, J.M.: On distortion and thickness of knots. In: Whittington, Sumners, Lodge (eds.) Topology and geometry in Polymer Science, pp. 67–78. IMA Volumes in Math. and its Appl. Vol. 103. Springer, Heidelberg (1998)

  31. Langer J. and Singer D.A. (1987). Curve-straightening in Riemannian manifolds. Ann. Global Anal. Geom. 5: 133–150

    Article  MATH  MathSciNet  Google Scholar 

  32. Léger J.C. (1999). Menger curvature and rectifiability. Ann. Math. 149(2): 831–869

    MATH  Google Scholar 

  33. Lin Y. and Mattila P. (2000). Menger curvature and C 1-regularity of fractals. Proc. AMS 129: 1755–1762

    Article  MathSciNet  Google Scholar 

  34. Litherland R.A., Simon J., Durumeric O.C. and Rawdon E. (1999). Thickness of knots. Topol. Appl. 91: 233–244

    Article  MATH  MathSciNet  Google Scholar 

  35. Marenduzzo D., Micheletti C., Seyed-allaei H., Trovato A. and Maritan A. (2005). Continuum model for polymers with finite thickness. J. Phys. A: Math. Gen. 38: L277–L283

    Article  MathSciNet  Google Scholar 

  36. O’Hara J. (1991). Energy of a knot. Topology 30: 241–247

    Article  MATH  MathSciNet  Google Scholar 

  37. O’Hara, J.: Energy of knots and conformal geometry. Ser. of Knots and Everything vol. 33. World Scientific, River Edge (2003)

  38. Rawdon E. and Simon J. (2002). Möbius energy of thick knots. Topol. Appl. 125: 97–109

    Article  MATH  MathSciNet  Google Scholar 

  39. Reiter, P.: Knotenenergien. Diplom thesis, Univ. Bonn (2004)

  40. Reiter, P.: All curves in a C 1-neighbourhood of a given embedded curve are isotopic. Preprint Nr. 4, Institut für Mathematik, RWTH Aachen 2005, see http://www.instmath.rwth-aachen.de/ → preprints

  41. Schuricht F. and von der Mosel H. (2003). Global curvature for rectifiable loops. Math. Z. 243: 37–77

    MATH  MathSciNet  Google Scholar 

  42. Schuricht F. and von der Mosel H. (2003). Euler-Lagrange equations for nonlinearly elastic rods with self-contact. Arch. Rat. Mech. Anal. 168: 35–82

    Article  MATH  MathSciNet  Google Scholar 

  43. Schuricht F. and von der Mosel H. (2004). Characterization of ideal knots. Calc. Var. Partial Diff. Equ. 19: 281–305

    Article  MathSciNet  Google Scholar 

  44. Simon J. (1994). Energy functions for polygonal knots. J. Knot Theory Ramifications 3: 299–320

    Article  MATH  MathSciNet  Google Scholar 

  45. Smutny, J.: Global radii of curvature, and the biarc approximation of space curves: In pursuit of ideal knot shapes. Ph.D. thesis no. 2981, EPFL Lausanne (2004)

  46. Stein E.M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton

    MATH  Google Scholar 

  47. Strzelecki, P., von der Mosel, H.: On a mathematical model for thick surfaces. In: Calvo, Millett, Rawdon, Stasiak (eds.) Physical and Numerical Models in Knot Theory, pp. 547–564. Ser. on Knots and Everything, vol. 36. World Scientific, Singapore (2005)

  48. Strzelecki P. and von der Mosel H. (2006). Global curvature for surfaces and area minimization under a thickness constraint. Calc. Var. 25: 431–467

    Article  MATH  MathSciNet  Google Scholar 

  49. Strzelecki, P., von der Mosel, H.: On rectifiable curves with L p-bounds on global curvature: self-avoidance, regularity, and minimizing knots. Preprint no. 6, Institut für Mathematik, RWTH Aachen, version of March 2006, see http://www.instmath.rwth-aachen.de/,˜heiko/lpbounds.pdf

  50. von der Mosel H. (1998). Minimizing the elastic energy of knots. Asymp. Anal. 18: 49–62

    MATH  MathSciNet  Google Scholar 

  51. von der Mosel H. (1999). Elastic knots in Euclidean 3-space. Ann. Inst. H. Poincaré Anal. Non Linéaire 16: 137–166

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Instytut Matematyki, Uniwersytet Warszawski, ul. Banacha 2, 02-097, Warsaw, Poland

    Paweł Strzelecki

  2. Institut für Mathematik, RWTH Aachen, Templergraben 55, 52062, Aachen, Germany

    Heiko von der Mosel

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  1. Paweł Strzelecki
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  2. Heiko von der Mosel
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Correspondence to Heiko von der Mosel.

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Strzelecki, P., von der Mosel, H. On rectifiable curves with L p-bounds on global curvature: self-avoidance, regularity, and minimizing knots. Math. Z. 257, 107–130 (2007). https://doi.org/10.1007/s00209-007-0117-4

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