Scale-invariant tangent-point energies for knots

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