Archive for Rational Mechanics and Analysis

, Volume 225, Issue 1, pp 89–139

The Elastic Trefoil is the Doubly Covered Circle

  • Henryk Gerlach
  • Philipp Reiter
  • Heiko von der Mosel

DOI: 10.1007/s00205-017-1100-9

Cite this article as:
Gerlach, H., Reiter, P. & von der Mosel, H. Arch Rational Mech Anal (2017) 225: 89. doi:10.1007/s00205-017-1100-9


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.

Funding information

Funder NameGrant NumberFunding Note
    • MO966/7-1
    • RE3930/1-1

    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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