Archive for Rational Mechanics and Analysis

, Volume 225, Issue 1, pp 89–139 | Cite as

The Elastic Trefoil is the Doubly Covered Circle

  • Henryk Gerlach
  • Philipp Reiter
  • Heiko von der Mosel


To describe the behavior of knotted loops of springy wire with an elementary mathematical model we minimize the integral of squared curvature, \({E = \int \varkappa^2}\), together with a small multiple of ropelength \({\mathcal{R}}\) = length/thickness in order to penalize selfintersection. Our main objective is to characterize all limit configurations of energy minimizers of the total energy \({E_{\vartheta} \equiv E + \vartheta \mathcal{R}}\) as \({\vartheta}\) tends to zero. For short, these limit configurations will be referred to as elastic knots. The elastic unknot turns out to be the once covered circle with squared curvature energy \({(2\pi)^2}\). For all (non-trivial) knot classes for which the natural lower bound \({(4\pi)^2}\) on E is sharp, the respective elastic knot is the doubly covered circle. We also derive a new characterization of two-bridge torus knots in terms of E, proving that the only knot classes for which the lower bound \({(4\pi)^2}\) on E is sharp are the \({(2,b)}\)-torus knots for odd b with \({|b| \ge 3}\) (containing the trefoil knot class). In particular, the elastic trefoil knot is the doubly covered circle.


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  1. 1.
    Antman, S.S.: Nonlinear problems of elasticity, Applied Mathematical Sciences, vol. 107, second edn. Springer, New York 2005Google Scholar
  2. 2.
    Artin, E.: Theorie der Zöpfe. Abh. Math. Sem. Univ. Hamburg 4(1), 47–72 1925. doi:10.1007/BF02950718
  3. 3.
    Avvakumov, S., Sossinsky, A.: On the normal form of knots. Russ. J. Math. Phys. 21(4), 421–429 2014. doi:10.1134/S1061920814040013.
  4. 4.
    Blatt, S.: Note on continuously differentiable isotopies. Report 34, Institute for Mathematics, RWTH Aachen 2009Google Scholar
  5. 5.
    Blatt, S., Reiter, Ph.: Stationary points of O’Hara’s knot energies. Manuscripta Math. 140(1-2), 29–50 2013. doi:10.1007/s00229-011-0528-8.
  6. 6.
    Blatt, S., Reiter, Ph.: How nice are critical knots? Regularity theory for knot energies. J. Phys. Conf. Ser. 544, 012,020 2014. doi:10.1088/1742-6596/544/1/012020
  7. 7.
    Blatt, S., Reiter, Ph.: Modeling repulsive forces on fibres via knot energies. Mol. Based Math. Biol. 2, 56–72 2014. doi:10.2478/mlbmb-2014-0004
  8. 8.
    Blatt, S., Reiter, Ph., Schikorra, A.: Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth. Trans. Am. Math. Soc. 368(9), 6391–6438 2016. doi:10.1090/tran/6603
  9. 9.
    Burde, G., Zieschang, H.: Knots, de Gruyter Studies in Mathematics, vol. 5, second edn. Walter de Gruyter & Co., Berlin 2003.
  10. 10.
    Cantarella J., Kusner R.B., Sullivan J.M.: On the minimum ropelength of knots and links. Invent. Math. 150(2), 257–286 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Carlen, M.: Computation and visualization of ideal knot shapes. PhD thesis, EPF Lausanne 2010.
  12. 12.
    Cha, J.C., Livingston, C.: Knotinfo: Table of knot invariants. Accessed September 30, 2016
  13. 13.
    Crowell, R.H., Fox, R.H.: Introduction to knot theory. Springer, New York 1977. Reprint of the 1963 original, Graduate Texts in Mathematics, No. 57
  14. 14.
    Denne, E.: Alternating Quadrisecants of Knots. ArXiv Mathematics e-prints 2005.
  15. 15.
    Diao, Y., Ernst, C., Janse van Rensburg, E.J.: Thicknesses of knots. Math. Proc. Camb. Philos. Soc. 126(2), 293–310 1999. doi:10.1017/S0305004198003338.
  16. 16.
    Fenchel, W.: Geschlossene Raumkurven mit vorgeschriebenem Tangentenbild. Jahresbericht der Deutschen Mathematiker-Vereinigung 39, 183–185 1930.
  17. 17.
    Gallotti, R., Pierre-Louis, O.: Stiff knots. Phys. Rev. E (3) 75(3), 031,801, 14 2007. doi:10.1103/PhysRevE.75.031801.
  18. 18.
    Gonzalez, O., Maddocks, J.H.: Global curvature, thickness, and the ideal shapes of knots. Proc. Natl. Acad. Sci. USA 96(9), 4769–4773 (electronic) 1999. doi:10.1073/pnas.96.9.4769.
  19. 19.
    Gonzalez, O., Maddocks, J.H., Schuricht, F., von der Mosel, H.: Global curvature and self-contact of nonlinearly elastic curves and rods. Calc. Var. Partial Differ. Equ. 14(1), 29–68 2002. doi:10.1007/s005260100089
  20. 20.
    He, Z.X.: The Euler-Lagrange equation and heat flow for the Möbius energy. Comm. Pure Appl. Math. 53(4), 399–431 2000. doi:10.1002/(SICI)1097-0312(200004)53:4<399::AID-CPA1>3.3.CO;2-4
  21. 21.
    Koch, R., Engelhardt, C.: Closed space curves of constant curvature consisting of arcs of circular helices. J. Geom. Graph. 2(1), 17–31 1998.
  22. 22.
    Langer, J., Singer, D.A.: Knotted elastic curves in \({{\bf R}^3}\). J. London Math. Soc. (2) 30(3), 512–520 1984. doi:10.1112/jlms/s2-30.3.512.
  23. 23.
    Langer, J., Singer, D.A.: Curve straightening and a minimax argument for closed elastic curves. Topology 24(1), 75–88 1985. doi:10.1016/0040-9383(85)90046-1.
  24. 24.
    Langer, J., Singer, D.A.: Lagrangian aspects of the Kirchhoff elastic rod. SIAM Rev. 38(4), 605–618 1996. doi:10.1137/S0036144593253290.
  25. 25.
    Lin, C.C., Schwetlick, H.R.: On a flow to untangle elastic knots. Calc. Var. Partial Differential Equations 39(3–4), 621–647 2010. doi:10.1007/s00526-010-0328-0.
  26. 26.
    Litherland, R.A., Simon, J.K., Durumeric, O.C., Rawdon, E.J.: Thickness of knots. Topol. Appl. 91(3), 233–244 1999. doi:10.1016/S0166-8641(97)00210-1.
  27. 27.
    McAtee Ganatra, J.M.: Knots of constant curvature. J. Knot Theory Ramifications 16(4), 461–470 2007. doi:10.1142/S0218216507005348.
  28. 28.
    Milnor, J.W.: On the total curvature of knots. Ann. of Math. (2) 52, 248–257 1950. doi:10.2307/1969467
  29. 29.
    O’Hara, J.: Energy of a knot. Topology 30(2), 241–247 1991. doi:10.1016/0040-9383(91)90010-2
  30. 30.
    O’Hara, J.: Energy of knots and conformal geometry, Series on Knots and Everything, vol. 33. World Scientific Publishing Co. Inc., River Edge, NJ 2003Google Scholar
  31. 31.
    Reiter, Ph.: All curves in a \({C^1}\)-neighbourhood of a given embedded curve are isotopic. Report 4, Institute for Mathematics, RWTH Aachen 2005Google Scholar
  32. 32.
    Reiter, Ph.: Regularity theory for the Möbius energy. Commun. Pure Appl. Anal. 9(5), 1463–1471 2010. doi:10.3934/cpaa.2010.9.1463.
  33. 33.
    Reiter, Ph.: Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family \({E^{(\alpha)}, \alpha\in[2,3)}\). Math. Nachr. 285(7), 889–913 2012. doi:10.1002/mana.201000090.
  34. 34.
    Rolfsen, D.: Knots and links. Publish or Perish, Inc., Berkeley, Calif. 1976. Mathematics Lecture Series, No. 7
  35. 35.
    Schubert, H.: Über eine numerische Knoteninvariante. Math. Z. 61, 245–288 1954. doi:10.1007/BF01181346.
  36. 36.
    Schur, A.: Über die Schwarzsche Extremaleigenschaft des Kreises unter den Kurven konstanter Krümmung. Math. Ann. 83(1–2), 143–148 1921. doi:10.1007/BF01464234.
  37. 37.
    Schuricht, F., von der Mosel, H.: Euler–Lagrange equations for nonlinearly elastic rods with self-contact. Arch. Ration. Mech. Anal. 168(1), 35–82 2003. doi:10.1007/s00205-003-0253-x.
  38. 38.
    Schuricht, F., von der Mosel, H.: Global curvature for rectifiable loops. Math. Z. 243(1), 37–77 2003. doi:10.1007/s00209-002-0448-0.
  39. 39.
    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(1), 107–130 2007. doi:10.1007/s00209-007-0117-4.
  40. 40.
    Strzelecki, P., von der Mosel, H.: Menger curvature as a knot energy. Phys. Rep. 530, 257–290 2013. doi:10.1016/j.physrep.2013.05.003
  41. 41.
    Strzelecki, P., von der Mosel, H.: How averaged menger curvatures control regularity and topology of curves and surfaces. In: Knotted, Linked and Tangled Flux in Quantum and Classical Systems, Journal of Physics Conference Series. IoP, Cambridge 2014. doi:10.1088/1742-6596/544/1/012018
  42. 42.
    Truesdell, C.: The rational mechanics of flexible or elastic bodies. 1638–1788. Leonhardi Euleri. Opera omnia. Ser. 2: Opera mechanica et astronomica. Vol. XI, sect. altera. Zürich: Orell Füssli, 435 p. 1960Google Scholar
  43. 43.
    von der Mosel, H.: Minimizing the elastic energy of knots. Asymptot. Anal.18(1–2), 49–65 1998.
  44. 44.
    von der Mosel, H.: Elastic knots in Euclidean 3-space. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(2), 137–166 1999. doi:10.1016/S0294-1449(99)80010-9.

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.EPF LausanneLausanneSwitzerland
  2. 2.Fakultät für MathematikUniversität Duisburg–EssenEssenGermany
  3. 3.Institut für MathematikRWTH Aachen UniversityAachenGermany

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