The Study of Ropelength and Knot Energies

jason Cantarella和Joseph fu该项目将研究结的几何形状，这是“绳长”的关键配置，其定义为其长度除以其最大嵌入管邻域半径的商。由于长度泛函不是光滑的，所以没有经典的欧拉-拉格朗日方程来描述最小值。项目666t的主要目的是提供欧拉-拉格朗日方程的类比：一个“平衡准则”，它表明曲线是长度临界的当且仅当核心曲线的长度的第一个变化可以被作用在周围管上的自接触力系统平衡。用来证明平衡准则的方法（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替代定理的一个版本）保证了在其他数学领域的丰富应用，例如凸曲线的研究，以及“结能”。具有圆形横截面的绳子的自然模型是一个由半径固定的不自相交的管子包围的空间曲线。如果这样一根绳子打一个结，并把结拉紧，得到的曲线是曲线“绳长”的关键配置，它被定义为曲线长度除以管的半径。该项目利用框架理论的思想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，并给出结构上的外部载荷如何分解为不同结构元素的张力和压缩的精确描述。该项目将紧结中的张力视为对绳子的每条曲线内部施加的力，目的是表明如果这种力可以分解成作用在管表面的自接触力系统，那么这样的曲线是紧的。这种“平衡标准”产生了令人惊讶的结果。想象一根绳子水平伸展，就像晾衣绳。如果另一根绳子穿过这条线，并向下拉，直到这对绳子拉紧，绳子就会形成四个笔直的部分，连接到一个中心区域，在那里，绳子缠绕在一起。人们会期望绳索在整个弯曲的扣环中保持接触，每个弯道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，两股之间总是有一个小的间隙。这个模型已经在分子生物学中被用来描述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该模型在物理学、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.