On some knot energies involving Menger curvature

TL;DR

This paper studies knot energies derived from Menger curvature, showing they promote self-avoidance, regularity, and often uniquely minimize to circles, with implications for knot complexity measures.

Contribution

It introduces a family of knot energies based on Menger curvature that ensure self-avoidance, regularity, and unique minimization properties, advancing understanding of knot geometry.

Findings

All energies are charge, minimizable, tight, and strong.

Most energies distinguish knots from unknots.

Some energies are uniquely minimized by circles.

Abstract

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. All of these energies turn out to be charge, minimizable in given isotopy classes, tight and strong. Almost all distinguish between knots and unknots, and some of them can be shown to be uniquely minimized by round circles. Bounds on the stick number and the average crossing number, some non-trivial global lower bounds, and unique minimization by circles upon compaction complete the picture.