Characterization of ideal knots

Abstract

We present a characterization of ideal knots, i.e., of closed knotted curves of prescribed thickness with minimal length, where we use the notion of global curvature for the definition of thickness. We show with variational methods that for an ideal knot $\gamma$ , the normal vector $\gamma"(s)$ at a curve point $\gamma(s)$ is given by the integral over all vectors $\gamma(\tau)-\gamma(s)$ against a Radon measure, where $\vert\gamma(\tau)-\gamma(s)\vert/2$ realizes the given thickness. As geometric consequences we obtain in particular, that points without contact lie on straight segments of $\gamma$ , and for points $\gamma(s)$ with exactly one contact point $\gamma(\tau)$ we have that $\gamma"(s)$ points exactly into the direction of $\gamma(\tau) -\gamma(s).$ Moreover, isolated contact points lie on straight segments of $\gamma$ , and curved arcs of $\gamma$ consist of contact points only, all realizing the prescribed thickness with constant (maximal) global curvature.