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
$E_{\rm b}(\mathbf{x })$
.
