On Thickness and Packing Density for Knots and Links

Total Curvature and Packing of Knots

We establish a new fundamental relationship between total curvature of knots and crossing number. If K is a smooth knot in 3-space, R the cross-section radius of a uniform tube neighborhood of K, L the arclength of K, and k the total curvature of K, then (up to a coefficient independent of K), crossing number of K < (k)(L/R). There are families of knots whose crossing numbers grow faster than either k or L/R separately. For example, the knots whose crossing numbers grow with the (4/3)-power of ropelength must have total curvature growing arbitrarily large as well.

On Distortion and Thickness of Knots

What length of rope (of given diameter) is required to tie a particular knot? To answer this question, we define some new notions of thickness for a space curve, one based on Gromov's distortion, and another generalizing the thickness of Litherland, Simon et al. We prove a basic inequality between these thickness measures, and show that the distortion thickness is upper semi-continuous in the C^0 topology, suggesting that shortest curves of thickness 1 should exist in each knot class.

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。

Local Structure of Ideal Shapes of Knots, II, Constant Curvature Case

The thickness, NIR(K) of a knot or link K is defined to be the radius of the largest solid tube one can put around the curve without any self intersections, which is also known as the normal injectivity radius of K. For C^{1,1} curves K, NIR(K)=min{(1/2)DCSC(K),(1/(supkappa(K))))}, where kappa(K) is the generalized curvature, and the double critical self distance DCSD(K) is the shortest length of the segments perpendicular to K at both end points. The knots and links in ideal shapes (or tight knots or links) belong to the minima of ropelength = length/thickness within a fixed isotopy class. In…

Topological and physical knot theory are distinct

Physical knots and links are one-dimensional submanifolds of R^3 with fixed length and thickness. We show that isotopy classes in this category can differ from those of classical knot and link theory. In particular we exhibit a Gordian Split Link, a two component link that is split in the classical theory but cannot be split with a physical isotopy.

关于纽结与链环的最小绳长

On the Minimum Ropelength of Knots and Links

该研究证明了任意纽结或链环类型中绳长极小值子的存在性与正则性，改进了非平凡纽结绳长下界，还为小型纽结与链环建立了含紧界的新绳长界。