Scale-invariant tangent-point energies for knots

Abstract

We investigate minimizers and critical points for scale-invariant tangent-point energies ${\mathrm{TP}}^{p,q}$ of closed curves. We show that (a) minimizing sequences in ambient isotopy classes converge to locally critical embeddings at all but finitely many points and (b) locally critical embeddings are regular. Technically, the convergence theory (a) is based on a gap estimate for fractional Sobolev spaces with respect to the tangent-point energy. The regularity theory (b) is based on constructing a new energy ${\mathcal{E}}^{p,q}$ and proving that the derivative ${\gamma }^{\prime }$ of a parametrization of a ${\mathrm{TP}}^{p,q}$-critical curve $\gamma$ induces a critical map with respect to ${\mathcal{E}}^{p,q}$ acting on torus-to-sphere maps.