Knot Complexity and the Structure of Polygonal Knot Space

结复杂度与多边形结空间的结构

• 批准号: 0074315
• 负责人: Eric Rawdon
• 金额: $ 7.32万
• 依托单位: Chatham College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-09-01 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0074315&HistoricalAwards=false
• 关键词: Knot; Complexity; Structure; Polygonal; Space

The investigator studies connections between knot complexityand polygonal knot spaces, and develops effective methods toquantify and and characterize knots. The project involvescomputation and software development as well as analysis andexperiments. With colleagues and students, the investigatorexplores relationships between various experimental measurementsof complexity of knots in physical materials, such as DNA andpolymers, and mathematical characterizations of knot complexity.Previously defined functions, such as energies and rope-length,are compared to new quantities, such as measurements of theconvex hull and of a "smallest" box containing the knot, tocapture various spatial characteristics. These quantitiespredict the types of knots that are encountered as one movesthrough knot space. They also are used to understand changesthat occur in small and large-scale knotting in polygonal knotspace as a result of perturbations. From DNA replication to unraveling one's garden hose,knotting and tangling are a part of many physical systems. Someknots are easier to tie (i.e. less complex), and thus more likelyto occur in these situations. How does one quantify thecomplexity of a knot? What measurable attributes fully explainthe complexity of a mathematical knot (i.e. a closed loop inspace)? Mathematicians have defined several functions, called"knot energies" that quantify the "tangledness" of knots.Simultaneously, scientists have completed physical experiments onknots made of real materials, such as DNA and polymers, thatdetermine other measures of complexity. To what extent are thetheoretical and experimental quantities related? Are thequantities delivering the same information or does each numberreveal something different about the knot? In particular, canone use these functions to create more realistic physical modelsof DNA? In this project, the investigator, colleagues, andstudents explore the quantification of knot complexity and itsrelation to spaces of polygonal knots by integrating theory withcomputer simulation. Previously defined theoretical measures,such as energies and rope-length, are compared to new quantities,such as the surface area and volume of the convex hull, tocapture various spatial characteristics related to the knot.Physical experiments and computer simulations are performed andstatistical analysis applied to understand their interrelations.These quantities also predict the types of knots that areencountered as one moves through polygonal knot space andexplains changes that occur in small and large-scale knotting asa result of perturbations. This provides scientists with abetter understanding of the mathematical models that arecurrently employed and suggest refinements to improve thesemodels.

研究人员研究了节点复杂性和多边形节点空间之间的联系，并发展了量化和刻画节点的有效方法。该项目包括计算和软件开发以及分析和实验。与同事和学生一起，研究人员探索了物理材料(如DNA和聚合物)中节点复杂性的各种实验测量与节点复杂性的数学特征之间的关系。将先前定义的函数，如能量和绳长，与新的量进行比较，如凸壳的测量和包含节点的“最小”盒子的测量，以捕捉各种空间特征。这些数量预测了一个人在结空间中移动时遇到的结的类型。它们也被用来理解多边形纽结空间中由于扰动而发生的小规模和大规模纽结的变化。从DNA复制到解开你的花园软管，打结和缠绕是许多物理系统的一部分。有些结更容易打结(即不那么复杂)，因此更有可能出现在这些情况下。如何量化一个结的复杂程度呢？什么可测量的属性完全解释了数学结(即空间中的闭合环)的复杂性？数学家们定义了几个函数，称为“结能量”，用来量化结的“纠缠度”。与此同时，科学家们已经完成了由真实材料(如DNA和聚合物)制成的结的物理实验，这些实验确定了其他复杂程度的衡量标准。理论量和实验量在多大程度上相关？这些数量传递的信息是相同的，还是每个数字都揭示了关于这个结的一些不同的东西？特别是，人们能否使用这些功能来创建更逼真的DNA物理模型？在这个项目中，研究人员、同事和学生通过将理论与计算机模拟相结合来探索节点复杂性的量化及其与多边形节点空间的关系。将以前定义的理论测量，如能量和绳长，与新的量，如凸壳的表面积和体积进行比较，以捕捉与结相关的各种空间特征。进行物理实验和计算机模拟，并应用统计分析来了解它们之间的相互关系。这些量还预测当一个人在多边形结空间中移动时遇到的结的类型，并解释由于扰动而发生的小的和大范围的打结的变化。这让科学家们更好地理解了目前常用的数学模型，并提出了改进这些模型的建议。