Mathematical Sciences: Low Dimensional Manifolds and Knot Theory

数学科学：低维流形和结理论

• 批准号: 9626550
• 负责人: Cameron Gordon
• 金额: $ 16.14万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626550&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Low; Dimensional; Manifolds

9626550 Gordon Cameron Gordon will continue to investigate Dehn surgery on knots. The main goal is further to circumscribe the various exceptional (non-hyperbolic) surgeries on hyperbolic knots. The known bounds on the number and nature of such surgeries are now quite sharp in many cases, and it seems that it may eventually be possible (at least modulo the Geometrization Conjecture of Thurston) to give an essentially complete description of all exceptional surgeries. The methods used will be based largely on the combinatorial- topological techniques developed by Gordon and John Luecke in their previous work on Dehn surgery, such as the proof of the Knot Complement Conjecture. This project deals with knot theory and 3-dimensional topology, the general goal of the latter being to understand the structure of 3-dimensional manifolds. These are objects that are locally like ordinary 3-dimensional Euclidean space but whose global structure may be quite complicated. Since we live in a 3-manifold, one might say that 3-dimensional topology aims to describe what the mathematical possibilities are for our spatial universe. This aim is still not realized, although there is tantalizing evidence that such a description might ultimately be possible. One important aspect of 3-dimensional topology is the theory of knots---a knot being a closed loop embedded somehow in ordinary 3-dimensional space. On the one hand, results about 3-manifolds often give information about knots as special cases, while on the other hand, a wide variety of mathematical methods can be applied to the study of knots, leading in turn to new information about 3-manifolds. (Recently, deep connections between 3-dimensional topology and quantum field theory were discovered in this way.) The main vehicle by which knot theory relates to the general theory of 3-manifolds is Dehn surgery, and this is the focus of the current project. In Dehn surgery, a solid tube around a knot is removed and "s ewn back" differently, giving a new 3-manifold. If one allows links (i.e., several loops linked together) as well as knots, then every 3-manifold can be obtained in this way, so a sufficiently good understanding of this construction would have important implications for the general theory of 3-manifolds. ***

9626550戈登·卡梅伦·戈登将继续调查德恩的绳结手术。主要目标是进一步限制各种特殊的(非双曲的)双曲纽结手术。在许多情况下，此类手术的数量和性质的已知界限现在相当尖锐，似乎最终可能(至少以瑟斯顿的几何化猜想为模)给出所有特殊手术的基本上完整的描述。所使用的方法将主要基于Gordon和John Luecke在他们之前关于Dehn手术的工作中开发的组合拓扑技术，例如证明纽结互补猜想。这个项目涉及纽结理论和三维拓扑学，后者的总体目标是了解三维流形的结构。这些物体局部类似于普通的三维欧几里德空间，但其全局结构可能相当复杂。由于我们生活在一个三维流形中，人们可能会说，三维拓扑学的目的是描述我们的空间宇宙的数学可能性。这一目标仍然没有实现，尽管有诱人的证据表明，这样的描述最终可能是可能的。三维拓扑学的一个重要方面是纽结理论-纽结是以某种方式嵌入普通三维空间的闭合环。一方面，关于三维流形的结果往往是关于纽结的特例，另一方面，各种各样的数学方法可以应用于纽结的研究，从而导致关于三维流形的新的信息。(最近，通过这种方式发现了三维拓扑和量子场论之间的深层次联系。)纽结理论与三维流形的一般理论相联系的主要载体是Dehn手术，这也是当前项目的重点。在Dehn手术中，一个结周围的实心管子被移除，并以不同的方式“S EWN背部”，给出了一个新的3-歧管。如果允许链接(即，几个环链接在一起)和纽结，那么每个3-流形都可以通过这种方式获得，因此对这种结构的足够好的理解将对3-流形的一般理论具有重要的意义。***