Minimizing the Elastic Energy of Knots

We consider the problem of minimizing the bending energy Eb = R 2 ds on isotopy classes of closed curves in IR 3 to model the elastic behaviour of knotted loops of springy wire. A potential of Coulomb type with a small factor ` 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 x ` under a given topological knot type for each ` ? 0 . Moreover, allowing smaller and smaller thickness ( ` & 0 ) and looking at a subsequence of the corresponding minimizers x ` , we obtain a generalized minimizer x of the bending energy Eb as a limit. It turns out that x is the once covered circle, if one considers the class of unknotted loops in IR 3 . In nontrivial knot classes, however, x must have double points, whose multiplicity and position on the curve is controlled by the value of the bending energy Eb(x) . AMS Classification: 49Q10, 53A04, 57R52 1 Introduction Physical..