RUI: Characterizing Energy-Minimizing Knots

研究者、合作者和学生研究不同的结能量函数和它们的最佳配置。具体的主题包括：描述绳长极小值的形状；寻找不同能量函数的最优值之间的关系；研究多边形能量最优作为边数的函数确定弹性在拉紧结和有效包装中的作用；绳长极小值附近多边形结空间结构的探索确定不同多性体长度概念之间的关系。从宇宙弦的巨大结到DNA的微观缠结，打结和缠结是大自然的一部分。一种结的一个物理属性是它的最小绳长（即绳长）。打结所需的一英寸半径的完全灵活的绳的数量)。绳长这个概念已经被生物学家、物理学家和数学家用来预测螺旋DNA的排列、打结的DNA在凝胶中移动的速度，以及在不同构型的大群体中打结的平均形状。绳长最小化构象有什么特别之处，以至于它们可以预测真实物理结的行为？什么空间属性影响这种行为？此外，真实的物理材料有其固有的局限性，比如抗弯曲性。如何将不灵活性纳入大分子的数学模型？材料的柔韧性如何影响打结的方式以及在小空间内打结的长度？特别是，这些知识是否可以用于设计既坚固又防滑的打结材料？研究者、合作者和学生研究这些关于真实物理系统中打结和缠结的基本问题。这些研究使我们对纳米技术和生物技术中存在的缠结结构有了更深入的了解，并用真实的物理材料制作了更好的结模型。

Rawdon The investigator, collaborators, and students studydifferent knot energy functions and their optimal configurations.Specific topics include: characterizing the shape of ropelengthminima; finding relationships between the optima of differentenergy functions; studying polygonal energy optima as a functionof the number of edges; determining the role of flexibility inpulling knots tight and in packing them efficiently; exploringthe structure of polygonal knot space near ropelength minima;determining relationships between different notions of polygonalropelength. From the massive knots of cosmic strings to the microscopictangling of DNA, knotting and tangling are a part of nature. Onephysical attribute of a type of knot is its minimum ropelength(i.e. the amount of one-inch radius perfectly flexible ropeneeded to tie a configuration of the given type of knot). Theidea of ropelength has been used by biologists, physicists, andmathematicians to predict the packing of helical DNA, the speedat which knotted DNA moves through gel, and the average shape ofa knot within a large population of different configurations.What is so special about ropelength-minimizing conformations thatthey predict the behavior of real physical knots? What spatialattributes influence this behavior? Furthermore, real physicalmaterials have inherent limitations, such as resistance tobending. How does one incorporate inflexibility intomathematical models of macromolecules? How does the flexibilityof the material affect how the knot tightens and how long lengthspack into small spaces? In particular, can this knowledge beused to design knot-making materials that are both strong andresistant to slipping? The investigator, collaborators, andstudents study these fundamental questions about knotting andtangling in real physical systems. These studies lead to adeeper understanding of tangled structures present innanotechnology and biotechnology and to better models of knotsmade with real physical materials.