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 class. 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-Slobodeckij 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 Möbius energy thanks to its Möbius 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 Möbius invariance.

中文翻译：

---

关于奥哈拉能量的两点说明

提议由奥哈拉（Jun O'Hara）在1991年引入的奥哈拉能量是为了回答给定结类中什么是“好”图形的问题。奥哈拉能量的一个特性是，结的数字“越好”，则能量值越小。在本文中，我们讨论有关奥哈拉能量的两个主题。首先，我们稍微概括一下奥哈拉能量，并考虑其有限性的特征。Blatt在2012年使用Sobolev-Slobodeckij空间来考虑O'Hara能量的有限性，因此我们自然地考虑了该空间的推广。另一个基本问题是了解奥哈拉能量的极小值。在几篇论文中已经解决了这个问题，其中有些是基于数值计算的。在这个方向上 我们讨论O'Hara能量的离散化，并给出一些数值计算的示例。金·库斯纳（Kim-Kusner）于1993年考虑了奥哈拉能量中的一种特别的能量，因其莫比乌斯不变性而被称为莫比乌斯能量，而舒尔特人在2014年建立了收敛特性。由于论点不依赖于莫比乌斯不变性，因此我们一般采用他们的论点。