On the geometry of stiff knots

Interdisciplinary Physics

Abstract

We analyse the geometry of a thin knotted string with bending rigidity. Two types of geometric properties are investigated. First, following the approach of von der Mosel [H. von der Mosel, Asymptotic Anal. 18, 49 (1998)], we derive upper bounds for the multiplicity of crossings and braids. Then, using a general inequality for the length of 3D curves derived by Chakerian [G.D. Chakerian, Proc. of the American Math. Soc. 15, 886 (1964)], we analyze the size and confinement of a knot

PACS

46.70.Hg Membranes, rods, and strings 02.10.Kn Knot theory 47.57.Ng Polymers and polymer solutions 87.14.gk DNA 

References

  1. 1.
    L. Euler, Bousquet, Lausannae et Genevae 24, E65A (1744)Google Scholar
  2. 2.
    J. Langer, D.A. Singer, J. London Math. Soc. 30, 512 (1984)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    R. Gallotti, O. Pierre-Louis, Phys. Rev. E 75, 031801 (2007)CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    B. Audoly, N. Clauvelin, S. Neukirch, Phys. Rev. Lett. 99, 164301 (2007)CrossRefADSGoogle Scholar
  5. 5.
    E. Guitter, E. Orlandini, J. Phys. A 32 1359 (1999); R. Metzler, A. Hanke, P.G. Dommersnes, Y. Kantor, M. Kardar, Phys. Rev. Lett. 88, 188101 (2002)MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    P.G. Dommersnes, Y. Kantor, M. Kardar, Phys. Rev. E 66, 031802 (2002)CrossRefADSGoogle Scholar
  7. 7.
    X.R. Bao, H.J. Lee, S.R. Quake, Phys. Rev. Lett. 91 265506 (2003)CrossRefADSGoogle Scholar
  8. 8.
    A.Y. Grosberg, Y. Rabin, Phys. Rev. Lett. 99, 217801 (2007)CrossRefADSGoogle Scholar
  9. 9.
    J.W. Milnor, Ann. Math. 52, 248 (1949)CrossRefMathSciNetGoogle Scholar
  10. 10.
    H. von der Mosel, Asymptotic Anal. 18, 49 (1998)MATHADSGoogle Scholar
  11. 11.
    G.D. Chakerian, Proc. of the American Math. Soc. 15, 886 (1964)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    SA. Wasserman, NR. Cozzarelli, Science 232 951 (1986); D. Buck, E. Flapan, Knot Theory for Scientific Objects, OCAMI Studies (2007), p. 75CrossRefADSGoogle Scholar
  13. 13.
    J. O’Hara, Energy of knots and conformal geometry (World Scientific, Singapore, 2003); J. O’Hara, Ideal Knots (World Scientific, River Edge, NJ, 1998)MATHGoogle Scholar
  14. 14.
    G. Buck, E.J. Rawdon Phys. Rev. E 70, 011803 (2004)CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Laboratoire de Spectrométrie Physique, CNRS Univ. J. FourierSaint Martin d’HèresFrance
  2. 2.Rudolf Peierls Centre for Theoretical PhysicsOxfordUK

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