Physical Knot Theory

西蒙9706789 研究者与G教授合作。巴克，圣安瑟伦学院，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缠绕在一起。 虽然结通常是作为理想化的一维细丝来研究的，但这里的重点是“物理结”，即由真实的物理材料制成的结，从绳子到DNA或其他大的柔性分子。 这些被建模为数学结赋予物理属性，如自排斥能量或厚度。 物理纽结理论的两个目标是数学建模和帮助理解真实的物理系统，并使用物理启发的纽结复杂性的措施，开发新的方法结识别和分类。 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，理解临界点和多边形结的配置空间，理解调和结的参数化和拓扑性质之间的联系，将结能量思想扩展到其他结构，如图形，DNA环的凝胶电泳建模，并利用结能量来帮助理解物理现象，如自辐射管和打结涡流。 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？ 不同种类的绳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理解DNA环的凝胶电泳，因此更好地理解凝胶过程本身;这是最重要的B iotechnology技术之一，因此扎实的理解很重要。 该项目需要强大的计算和可视化，因此它将在这些领域培训学生并扩展技术。

Simon 9706789 The investigator collaborates with Prof. G. Buck, St. Anselm College, 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.