Minimizing the elastic energy of knots

Article type: Research Article

Authors: von der Mosel, Heiko

Affiliations: Mathematisches Institut, Universität Bonn, Beringstraße 4, 53115 Bonn, Germany E‐mail: heiko@johnson.math.uni‐bonn.de

Abstract: We consider the problem of minimizing the bending energy E_{\rm b}=\int\kappa^2\,{\rm d}s on isotopy classes of closed curves in \mathbb{R}^3 to model the elastic behaviour of knotted loops of springy wire. A potential of Coulomb type with a small factor \theta as a measure for the thickness of the wire is added to the elastic energy in order to preserve the isotopy class. With a direct method we show existence of minimizers \mathbf{x }^{\theta } under a given topological knot type for each \theta>0. Moreover, allowing smaller and smaller thickness (\theta\searrow0) and looking at a subsequence of the corresponding minimizers \mathbf{x }^{\theta}, we obtain a generalized minimizer \mathbf{x } of the bending energy E_{\rm b} as a limit. It turns out that \mathbf{x } is the once covered circle, if one considers the class of unknotted loops in {\Bbb R}^3. In nontrivial knot classes, however, \mathbf{x } must have double points, whose multiplicity and position on the curve is controlled by the value of the bending energy E_{\rm b}(\mathbf{x }).

Journal: Asymptotic Analysis, vol. 18, no. 1-2, pp. 49-65, 1998