On some knot energies involving Menger curvature

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.

31 pages, 4 figures; version 2 with minor changes and modifications

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Cited by in corpus (6)

• Self-repulsiveness of energies for closed submanifolds
• Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies
• Minimal $W^{s,\frac{n}{s}}$-harmonic maps in homotopy classes
• Symmetric critical knots for O'Hara's energies
• Banach gradient flows for various families of knot energies
• On the Analyticity of Critical Points of the Generalized Integral Menger Curvature in the Hilbert Case