扭结曲线的第二包壳

关于环面链节的壳数

On the hull numbers of torus links

R^3中曲线并的第n个外壳是具有以下性质的点集：任何通过该点的平面与曲线相交至少2n次。链路L的外壳数u（L）被定义为L的代表可以具有的非空外壳的最小数量。我们表明，环面链接的船体数量是小于预期的，但仍然很大。特别地，对于类型（p，p）的链路，壳数等于3 p/5，并且通常对于q>p的类型（p，q），壳数大于或等于p/2。

Knoto-ID: a tool to study the entanglement of open protein chains using the concept of knotoids

The backbone of most proteins forms an open curve. To study their entanglement, a common strategy consists in searching for the presence of knots in their backbones using topological invariants. However, this approach requires to close the curve into a loop, which alters the geometry of curve. Knoto-ID allows evaluating the entanglement of open curves without the need to close them, using the recent concept of knotoids which is a generalization of the classical knot theory to open curves. Knoto-ID can analyse the global topology of the full chain as well as the local topology by exhaustively s…

拓扑导向模拟揭示几何约束对打结蛋白质的影响

Topologically Directed Simulations Reveal the Impact of Geometric Constraints on Knotted Proteins

针对聚合物或丝状结构打结/解结模拟依赖随机过程的问题，提出拓扑引导方法确定最优路径，应用于切向球链模型并施加蛋白质几何约束，发现约束会增加扭结频率，揭示其对打结蛋白质的影响。

Quadrisecants and essential secants of knots: with applications to the geometry of knots

A quadrisecant line is one which intersects a curve in at least four points, while an essential secant captures something about the knottedness of a knot. This survey article gives a brief history of these ideas, and shows how they may be applied to questions about the geometry of a knot via the total curvature, ropelength and distortion of a knot.

The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti de Sitter geometry

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces c…

The length, width, and inradius of space curves

The width $w$ of a curve $γ$ in Euclidean space $R^n$ is the infimum of the distances between all pairs of parallel hyperplanes which bound $γ$, while its inradius $r$ is the supremum of the radii of all spheres which are contained in the convex hull of $γ$ and are disjoint from $γ$. We use a mixture of topological and integral geometric techniques, including an application of Borsuk Ulam theorem due to Wienholtz and Crofton's formulas, to obtain lower bounds on the length of $γ$ subject to constraints on $r$ and $w$. The special case of closed curves is also considered in each category. Our e…

映射类群的几何与刚性

Geometry and rigidity of mapping class groups

利用曲线复形的双曲性质，研究了映射类群MCG（S）的大尺度几何。我们证明了MCG（S）的任何自拟等距（除了少数零星的情形）都是离左乘有界距离的，从而得到了MCG（S）的拟等距刚性，即与MCG（S）拟等距的群几乎等于它（Hamenstadt用不同的方法证明了后一个定理）.作为我们的方法的一部分，我们得到了其他几个结构性的结果：树阶结构的描述的渐近锥面的MCG（S）;的曲线复投影映射从MCG（S）的本质子曲面的曲线复的产品的形象的特征;和建设的Sigma壳在MCG（S），一个类似的凸包。

Proteins analysed as virtual knots

Long, flexible physical filaments are naturally tangled and knotted, from macroscopic string down to long-chain molecules. The existence of knotting in a filament naturally affects its configuration and properties, and may be very stable or disappear rapidly under manipulation and interaction. Knotting has been previously identified in protein backbone chains, for which these mechanical constraints are of fundamental importance to their molecular functionality, despite their being open curves in which the knots are not mathematically well defined; knotting can only be identified by closing the…