Polygonal knot space near ropelength-minimized knots

Polygonal approximation and energy of smooth knots

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and the energies must be correctly regularized. We determine an explicit error bound between the energies in terms of the number of the edges of the polygon and the Ropelength of the smooth curve.

紧链接的紧界：T(Q,Q) torus链接的绳长

Tight Bounds for Tight Links: Ropelength of T(Q,Q) torus links

本文研究T(Q,Q) torus链接的绳长界，推导出更严格的下界，并展示通过几何优化可使绳长接近下界，为特定结类的绳长界研究提供了重要进展。

结型的几何密度与压缩半径

Geometric densities and compression radii of knot types

研究嵌入三维欧几里得空间中的闭合曲线的尺度不变几何量，关注其在固定结型下的优化行为。定义D-密度、D-压缩半径及打包比，探讨其在优化后的分离特性。

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。