The Distortion of a Knotted Curve

Quantum knots and knotted zeros

In this paper we show how to place Michael Berry's discovery of knotted zeros in the quantum states of hydrogen in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent wave function for hydrogen, as a map from three space to the complex plane and such that the inverse image of zero in the complex plane contains a knotted curve in three space. We show that for knots in three space this is a generic situation in that every smooth knot K in three space has a smooth classifying map f from three space to the complex plane such th…

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.

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…

Knot types of generalized Kirchhoff rods

Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-sectional inflation. Our main result gives explicit parameterizations for all periodic critical framed curves for this generalized functional. The main technical tool is a correspondence between the moduli space of shape similarity classes of closed framed curves and an infinite-dimen…

Alternating Quadrisecants of Knots

It is known that for every knotted curve in space, there is a line intersecting it in four places, a quadrisecant. Comparing the order of the four points along the line and knot we can distinguish three types of quadrisecants; the alternating ones have the most relevance for the geometry of a knot. In this paper we prove that every (nontrivial tame) knot has an alternating quadrisecant. This result had applications to the total curvature, second hull and ropelength of knots.

A Class of Curves In Every Knot Type Where Chords of High Distortion are Common

The distortion of a curve is the supremum, taken over distinct pairs of points of the curve, of the ratio of arclength to spatial distance between the points. Gromov asked in 1981 whether a curve in every knot type can be constructed with distortion less than a universal constant C. Answering Gromov's question seems to require the construction of lower bounds on the distortion of knots in terms of some topological invariant. We attempt to make such bounds easier to construct by showing that pairs of points with high distortion are very common on curves of minimum length in the set of curves in…

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。

Studies of global and local entanglements of individual protein chains using the concept of knotoids

We study here global and local entanglements of open protein chains by implementing the concept of knotoids. Knotoids have been introduced in 2012 by Vladimir Turaev as a generalization of knots in 3-dimensional space. More precisely, knotoids are diagrams representing projections of open curves in 3D space, in contrast to knot diagrams which represent projections of closed curves in 3D space. The intrinsic difference with classical knot theory is that the generalization provided by knotoids admits non-trivial topological entanglement of the open curves provided that their geometry is frozen a…

Seifert 面的外同调和结的扭曲

Knot distortion via curve combinatorics

研究证明了 torus 结的扭曲下界及 Seifert 面的外同调性质，揭示了结的几何特性与 monodromy 映射的关系。

在 $S^4$ 中通过边缘手术得到的打结曲面的可扩展映射类

Extendable mapping classes of knotted surfaces obtained by rim surgery in $S^4$

研究在 $S^4$ 中通过沿非分离曲线进行标准无扭边缘手术得到的打结曲面的可扩展映射类子群，并利用结群的子午线-经线刚性条件精确计算该子群。

平面类纽结的新量子不变量

New Quantum Invariants of Planar Knotoids

该研究将类纽结用于开曲线纽结建模，定义双标架平面类纽结并构造可多项式时间计算的新量子不变量，改进了交叉数≤5的平面类纽结的分类。

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…