Infinitesimal bending of knots and energy change

The elastic trefoil is the twice covered circle

We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy $E_{\text{bend}}=\intκ^2$ together with a small multiple of ropelength $\mathcal R=\text{length}/\text{thickness}$ in order to penalize selfintersection. Our main objective is to characterize elastic knots, i.e., all limit configurations of energy minimizers of the total energy $E_\vartheta:=E_{\text{bend}}+\vartheta\mathcal R$ as $\vartheta$ tends to zero. The elastic unknot turns out to be the round circle with bending energy $(2π)^2$. For all (non-trivial) knot classes fo…

弹性BB纽结是多重圆

Elastic BB knots are multifold circles

本文证明在广义Fáry-Milnor不等式下，单分量BB纽结类中的弹性纽结必为覆盖桥指数次的圆，通过比较曲线和变分方法简化分类。

Symmetric elastic knots

Minimizing the bending energy within knot classes leads to the concept of elastic knots which has been initiated in [von der Mosel, Asymptot. Anal. 1998]. Motivated by numerical experiments in arxiv:1804.02206 (doi:10.1090/mcom/3633) we prescribe dihedral symmetry and establish existence of dihedrally symmetric elastic knots for knot classes admitting this type of symmetry. Among other results we prove that the dihedral elastic trefoil is the union of two circles that form a (planar) figure-eight. We also discuss some generalizations and limitations regarding other symmetries and knot classes.

Legendrian 纽结在海森堡群中的 CR 不变能量

CR-invariant energy of Legendrian knots in the Heisenberg group

本文研究了海森堡群中 Legendrian 纽结的 CR 不变能量，通过正则化 Korányi 距离下的 -2 阶势能积分来定义能量函数，并展示了实数圆作为能量最小化者的特性，同时给出了海森堡群中的 Doyle--Schramm 余弦公式类比。

一类带正曲率和平点的二维曲面的非刚性

Nonrigidity of a class of two dimensional surfaces with positive curvature and planar points

构造了一类二维曲面的非平凡无穷小弯曲。本文所考虑的曲面是可定向的、紧的、有边界的、除了在许多平点处外处处具有正曲率的、第一同调群为零的曲面，从而得到了这类曲面的非刚性结果.

曲面翻译和折纸镶嵌中无伸长弯曲问题的完整闭式解

Complete closed-form solutions to the problem of inextensional bending for surfaces of translation and origami tessellations

本文提出曲面翻译和折纸镶嵌中无伸长弯曲问题的完整闭式解，揭示三种无限小无伸长变形模式及其对应的伸展、弯曲和扭曲效应。

Decomposition of generalized O'Hara's energies

O'Hara introduced several functionals as knot energies. One of them is the Möbius energy. We know its Möbius invariance from Doyle-Schramm's cosine formula. It is also known that the Möbius energy was decomposed into three components keeping the Möbius invariance. The first component of decomposition represents the extent of bending of the curves or knots, while the second one indicates the extent of twisting. The third one is an absolute constant. In this paper, we show a similar decomposition for generalized O'Hara energies. On the way to derive it, we obtain an analogue of the cosine formul…