Research at Undergraduate Institutions, Collaborative Research: Physical Knot Theory

研究者与爱荷华大学J. Simon教授合作，研究结及其应用。结是空间中的闭合环，可能以复杂而重要的方式纠缠在一起。虽然结通常被研究为理想化的一维细丝，但这里的重点是“物理结”，由真正的物理材料制成的结，从绳子到DNA或其他大的柔性分子。这些被建模为具有物理性质的数学结，如自排斥能量或厚度。物理结理论的两个目标是建立数学模型并帮助理解真实的物理系统，以及使用物理上的结复杂性度量来开发结识别和分类的新方法。该项目的具体目标包括：确定结复杂性的不同度量之间的精确关系，理解关键点和多边形结的构型空间，理解参数化和谐波结的拓扑特性之间的联系，将结能量的思想扩展到其他结构，如图，DNA环的凝胶电泳建模，以及使用结能量来帮助理解物理现象，如自辐射管和结涡。打结和缠结发生在科学研究的每一个尺度上，从微观的DNA环到日常的绳子，再到日冕中缠结的磁场环。研究人员和他们的学生将研究在所有物理系统中出现的基本问题：结和缠结是如何产生的？物理系统的什么数学性质导致结变得简化或完全解开？不同类型结的数学性质如何影响它们的物理行为？一个重点任务是更好地了解DNA环的凝胶电泳，从而了解凝胶过程本身；这是最重要的生物技术之一，因此对其有深入的了解非常重要。该项目需要强大的计算和可视化能力，因此它将在这些领域训练学生，并扩展技术。

Buck 9706865 The investigator collaborates with Prof. J. Simon, University of Iowa, to study knots and their applications. Knots are closed loops in space that may be tangled in complicated and essential ways. While knots usually have been studied as idealized one dimensional filaments, the focus here is on "physical knots," knots made of real physical stuff, from rope to DNA or other large flexible molecules. These are modeled as mathematical knots endowed with physical like properties such as self repelling energy or thickness. The twin goals of physical knot theory are to mathematically model and help understand the real physical systems, and to use the physically inspired measures of knot-complexity to develop novel methods for knot recognition and classification. The specific targets of this project include: determining precise relations between different measures of knot complexity, understanding critical points and the configuration space of polygonal knots, understanding the connections between parametrizations and topological properties of harmonic knots, extending knot energy ideas to other structures such as graphs, modeling gel electrophoresis of DNA loops, and using knot energies to help understand physics phenomena such as self-radiating tubes and knotted vortices. Knotting and tangling happen at every scale studied by science, from microscopic DNA loops to everyday rope to tangled magnetic field loops in the solar corona. The investigators and their students will study problems that are fundamental and arise in all the physical systems: How are knots and tangles created? What mathematical properties of the physical systems lead to knots becoming simplified or completely untangled? How do the mathematical properties of different kinds of knots influence their physical behavior? One focus task is to gain a better understanding of gel electrophoresis of DNA loops, hence of the gel process itself; this is one of the most important b iotechnology techniques, so a solid understanding is important. The project requires powerful computing and visualization, so it will train students, as well as stretch the technology, in these areas.