Self-contact Sets for 50 Tightly Knotted and Linked Tubes

Knotted structures in high-energy Beltrami fields on the torus and the sphere

Let S be a finite union of (pairwise disjoint but possibly knotted and linked) closed curves and tubes in the round sphere S^3 or in the flat torus T^3. In the case of the torus, S is further assumed to be contained in a contractible subset of T^3. In this paper we show that for any sufficiently large odd integer λthere exists a Beltrami field on S^3 or T^3 satisfying curl u = λu and with a collection of vortex lines and vortex tubes given by S, up to an ambient diffeomorphism.

基于约束梯度下降的绳结收紧方法

Knot Tightening By Constrained Gradient Descent

该研究提出约束梯度下降法求解固定厚度下近似长度最小的打结多边形，推导了相关理论准则，得到379个覆盖指定交叉数范围的素纽结与链环的绳长结果及自接触图谱，提升了纽结绳长精度并首次发表链环绳长数据。