The Study of Ropelength and Knot Energies

DMS-0204826 Jason Cantarella和Joseph Fu该项目将研究结的几何形状，这些结是“绳长”的关键配置，绳长被定义为它们的长度与它们最大嵌入管邻域半径的乘积。由于ropelength泛函是不光滑的，没有经典的Euler-Lagrange方程描述极小元。 项目666 t的主要目的是提供一个类似于欧拉-拉格朗日方程的“平衡准则”，它表明当且仅当核心曲线长度的第一变化可以通过作用在周围管上的自接触力系统来平衡时，曲线是ropelength critical。平衡判据的证明方法（将张拉整体理论的某些方面推广到一类“连续张拉整体”，使用Banach空间上线性算子的Farkas择一定理的一个版本）在数学的其他领域有着丰富的应用，例如凸曲线的研究，具有圆形横截面的绳索的自然模型是由固定半径的非自相交管包围的空间曲线。 如果这样的绳子系成一个结，并且这个结被拉紧，那么所得到的曲线对于曲线的“ropelength”来说是一个关键的配置，该曲线的“ropelength”被定义为曲线的长度除以管的半径的商。该项目利用框架理论的思想来研究长度临界曲线的几何形状。该理论研究结构的简单工程模型，并精确描述结构上的外部载荷如何分解为不同结构元件的拉伸和压缩。 该项目将紧结中的张力看作是对绳子每个曲线内侧施加的力，并旨在表明，如果这种力可以分解为作用在管表面上的自接触力系统，那么这种曲线就是紧的。这种“平衡准则”具有令人惊讶的结果。想象一根水平伸展的绳子，就像晾衣绳。如果另一根绳子穿过绳子，向下拉，直到绳子拉紧，绳子就形成了四个直的部分，连接到一个中心区域，在那里绳子相互缠绕。人们会期望绳子在整个弯曲的扣钩中保持接触，每个弯曲内侧的两个点彼此接触。然而，根据平衡标准，这是不可能发生的：在转弯的中心，两股线之间总是有一个小的间隙。这个模型已经在分子生物学中用于描述DNA链的行为;该项目提供的新的几何信息将有助于改进和扩展这个模型在物理学、生物学和工程学中的许多其他应用。

DMS-0204826Jason Cantarella and Joseph FuThe project will study the geometry of knots which are criticalconfigurations for "ropelength", which is defined to be the quotientof their length over the radius of their largest embedded tubularneighborhood. Since the ropelength functional is not smooth, there isno classical Euler-Lagrange equation describing the minimizers. Theprimary aim of the projec666t is to provide an analogy to theEuler-Lagrange equation: a "balance criterion", which shows that acurve is ropelength critical if and only if the first variation of thelength of the core curve can be balanced by a system of self-contactforces acting on the surrounding tube. The methods used to prove thebalance criterion (a generalization of certain aspects of tensegritytheory to a class of "continuous tensegrities", using a version of theFarkas alternative theorem for linear operators on Banach spaces)promise rich applications in other areas of mathematics, such as thestudy of convex curves, and of "knot energies".A natural model for a rope with a circular cross-section is a spacecurve surrounded by a non-self-intersecting tube of fixed radius. Ifsuch a rope is tied in a knot, and the knot is pulled tight, theresulting curve is a critical configuration for the "ropelength" ofthe curve, which is defined to be the quotient of the length of thecurve over the radius of the tube. The project studies the geometry ofropelength-critical curves using ideas from the theory of frameworks.This theory studies simple engineering models of structures, and givesa precise description of how external loads on a structure areresolved into tensions and compressions of different structuralelements. The project views the tension in a tight knot as exerting aforce directed towards the inside of each curve of the rope, and aimsto show that such a curve is tight if this force can be resolved intoa system of self-contact forces acting on the surface of the tube.This "balance criterion" has surprising consequences. Imagine a ropestretched horizontally, like a clothesline. If another rope is passedover the line, and pulled down until the pair is tight, the ropes formfour straight segments joined to a central region where the strandscurve around one another. One would expect the ropes to maintaincontact throughout this curved clasp, with the two points on theinside of each bend touching one another. However, according to thebalance criterion, this cannot happen: there is always a small gapbetween the two strands at the center of the turn. This model hasalready been used in molecular biology to describe the behavior of DNAstrands; the new geometric information provided by the project willhelp to refine and extend many other appications of this model inphysics, biology, and engineering.