Scale-invariant tangent-point energies for knots

TL;DR

This paper studies scale-invariant tangent-point energies for knots, proving convergence of minimizers and establishing regularity of critical points through new energy constructions and fractional Sobolev space estimates.

Contribution

It introduces a new energy functional and provides a convergence and regularity theory for critical points of tangent-point energies in knot theory.

Findings

Minimizing sequences converge to locally critical embeddings.

Locally critical embeddings exhibit regularity.

A new energy functional aids in regularity analysis.

Abstract

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.