Combinatorial and Geometric Problems in Knot Theory

结理论中的组合和几何问题

• 批准号: 9971244
• 负责人: Morwen Thistlethwaite
• 金额: $ 6.99万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971244&HistoricalAwards=false
• 关键词: Combinatorial; Geometric; Problems; Knot; Theory

Proposal: DMS-9971244PI: Morwen ThistlethwaiteAbstract: Thistlethwaite's research is in classical knot theory, the study of embeddings of smooth simple closed curves in the 3-sphere. The computer is an essential tool in his research, in that the problems he addresses are often suggested by careful observation of data. Thistlethwaite will continue to investigate the rate of growth of the number of knots, following his collaboration with C. Sundberg, where the exact growth exponent was determined for prime, alternating links. He will study optimal configurations of symmetric knots, and will investigate other geometric problems associated with knot complements, in particular hyperbolic knot complements containing essential4-punctured spheres. He will investigate the possibility that hyperbolic geometry might be used to provide a new, completely geometric proof of the Tait flyping conjecture. The knot tables will be extended well beyond the 1,701,936 knots currently listed, and these new tables will be used to search for interesting examples. Development will continue on the freely available software package "Knotscape", which provides a graphical interface to the knot tables and access to many invariants.Thistlethwaite's specialty is classical knot theory, a branch of 3-dimensional topology with origins in the nineteenth century. A knot is a closed curve in 3-dimensional space; one can imagine a knotted rope with its ends joined together. Two knots are considered to be equivalent if one can be deformed continuously to the other, and a knot is deemed to be trivial, or unknotted, if it is equivalent to a flat circle. Knot theory is a rich subject, as it interfaces with geometry, topology, algebra and combinatorics. In recent years it has aroused the interest of chemists and molecular biologists, particularlyin relation to the knotting of DNA. Knot theory abounds with problems which are simple to state, but hard to solve; for example, it is often difficult in practice to prove that two given knots are inequivalent, as it is hard to rule out the existence of some ingenious method of deforming one to the other. It can even be hard to decide whether a given knot is trivial. If a knot is laid down on a flat surface with as few crossovers as possible, the resulting number of crossovers is called the crossing-number of the knot. To date, Thistlethwaite has classified by computer the 1,701,936 knots of up to 16 crossings; this classification was confirmed by an independent tabulation carried out by J. Hoste and J. Weeks. As part of this project, he will extend the tables to 17 or 18 crossings, thereby expecting to find many new examples with exciting properties. He will continue to investigate the fundamental problem as to how fast the number of knots grows in relation to crossing-number. Thistlethwaite will use hyperbolic geometry, a form of non-Euclidean geometry, to investigate hidden symmetries of knots, and to determine the structure of alternating knots (a knot is alternating if it can be arranged so that the rope goes alternately over and under at successive crossings.) He will continue to develop the software package "Knotscape", which provides a graphical interface to the knot tables.

提案：DMS-9971244 PI：Morwen Thistlethwaite摘要：Thistlethwaite的研究是在经典的纽结理论中，研究光滑简单闭曲线在3-球面中的嵌入。 计算机是他研究的重要工具，因为他所解决的问题往往是通过仔细观察数据提出的。 Thistlethwaite将继续研究结的数量增长率，他与C。Sundberg，其中确切的增长指数是为素数，交替链接确定的。 他将研究最佳配置的对称结，并将调查其他几何问题与结补，特别是双曲结补包含essential 4穿孔领域。 他将调查的可能性，双曲几何可能被用来提供一个新的，完全几何证明泰特flyping猜想。 纽结表将远远超出目前列出的1，701，936个纽结，这些新表将用于搜索有趣的例子。 将继续开发免费提供的软件包“Knotscape”，它提供了一个图形界面的结表和访问许多invariants.Thistlethwaite的专业是经典的结理论，一个分支的三维拓扑与起源于十九世纪。 结是三维空间中的一条闭合曲线;人们可以想象一根两端连接在一起的打结的绳子。 两个结被认为是等效的，如果一个可以连续变形到另一个，一个结被认为是平凡的，或未打结的，如果它相当于一个平面圆。 纽结理论是一门内容丰富的学科，因为它与几何、拓扑、代数和组合学紧密相连。 近年来，它引起了化学家和分子生物学家的兴趣，特别是与DNA的打结有关。纽结理论中有很多问题，这些问题很容易表述，但很难解决;例如，在实践中通常很难证明两个给定的纽结是不等价的，因为很难排除存在某种巧妙的方法将一个变形为另一个。 甚至很难确定一个给定的结是否是微不足道的。 如果一个结被放置在一个平面上，尽可能少的交叉，交叉的数量被称为结的交叉数。 到目前为止，Thistlethwaite已经通过计算机对多达16个交叉点的1，701，936节进行了分类;这一分类得到了J. Hoste和J. Weeks进行的独立列表的证实。 作为该项目的一部分，他将把表格扩展到17或18个交叉点，从而期望找到许多具有令人兴奋的特性的新例子。 他将继续调查的基本问题，如何快速增长的数量结的交叉数。 Thistlethwaite将使用双曲几何，一种非欧几何的形式，来研究结的隐藏对称性，并确定交替结的结构（如果一个结可以被安排，使得绳子在连续的交叉点上交替地上下移动，那么它就是交替结）。 他将继续开发软件包“Knotscape”，该软件包为结表提供图形界面。