Knot Complexity and the Structure of Polygonal Knot Space

研究者研究了结复杂性和多边形结空间之间的联系，并开发了有效的方法来量化和表征结。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，研究了物理材料（如DNA和聚合物）中结复杂性的各种实验测量与结复杂性的数学表征之间的关系。先前定义的功能，如能量和绳长，与新的数量进行比较，如凸船体和包含结的“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。它们也被用来理解由于扰动而在多边形结空间中发生的小结和大结的变化。从DNA复制到解开花园软管，打结和缠结是许多物理系统的一部分。有些结更容易打（即不那么复杂），因此更有可能在这些情况下发生。如何量化一个结的复杂性？哪些可测量的属性可以完全解释数学结（即空间中的闭环）的复杂性？数学家已经定义了几个函数，称为“结能”，用来量化结的“缠结”。与此同时，科学家们已经完成了由真实材料（如DNA和聚合物）构成的结的物理实验，这些实验确定了复杂性的其他衡量标准。理论量和实验量在多大程度上相关？数量传递的信息是相同的，还是每个数字都揭示了结的不同之处？特别是，能否使用这些功能来创建更真实的DNA物理模型？在这个项目中，研究者、同事和学生通过将理论与计算机模拟相结合来探索结复杂性的量化及其与多边形结空间的关系。先前定义的理论测量，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的相互关系。这些量还可以预测当一个人在多边形结空间中移动时遇到的结的类型，并解释由于扰动而在小结和大结中发生的变化。这为科学家们提供了对目前使用的数学模型的更好理解，并提出了改进这些模型的建议。

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.