Recent topics on the O'Hara energies

TL;DR

This paper explores generalizations of O'Hara energies, characterizes their finiteness, and discusses discretization and numerical computation methods for understanding their minimizers, extending previous work on M"{o}bius energy.

Contribution

It introduces a generalized framework for O'Hara energies, analyzes their finiteness conditions, and applies discretization techniques for numerical minimization, broadening prior research.

Findings

Characterization of finiteness conditions for generalized O'Hara energies

Development of a discretization method for numerical analysis

Demonstration of convergence properties for the discretized energies

Abstract

The O'Hara energies, introduced by Jun O'Hara in 1991, were proposed to answer the question of what is a "good" figure in a given knot type. A property of the O'Hara energies is that the "better" the figure of a knot is, the less the energy value is. In this article, we discuss two topics on the O'Hara energies. First, we slightly generalize the O'Hara energies and consider a characterization of its finiteness. The finiteness of the O'Hara energies was considered by Blatt in 2012 who used the Sobolev-Slobodeckii space, and naturally we consider a generalization of this space. Another fundamental problem is to understand the minimizers of the O'Hara energies. This problem has been addressed in several papers, some of them based on numerical computations. In this direction, we discuss a discretization of the O'Hara energies and give some examples of numerical computations. Particular one of the O'Hara energies, called the M\"{o}bius energy thanks to its M\"{o}bius invariance, was considered by Kim-Kusner in 1993, and Scholtes in 2014 established convergence properties. We apply their argument in general since the argument does not rely on M\"{o}bius invariance.