Mathematical Sciences: Knots, 3-Manifolds and Thin Position

数学科学：结、3 流形和薄位置

• 批准号: 9704140
• 负责人: Abigail Thompson
• 金额: $ 8.74万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704140&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Knots; Manifolds; Thin

9704140 Thompson A fundamental question in 3-dimensional topology is the classification problem for 3-manifolds, that is, to find a complete list of 3-manifolds in which each space appears precisely once. The classification of 2-dimensional manifolds provides a model of the kind of classification sought for 3-manifolds. A crucial step in the classification problem was completed by J. H. Rubinstein in 1992, when he described an algorithm to decide if a given 3-manifold was homeomorphic to the 3-sphere, the "simplest" 3-manifold. His arguments used minimal surface theory in the context of a triangulated 3-manifold. There is a deep connection between this type of minimal surface theory and the notion of thin position for knots in the 3-sphere, an idea introduced by D. Gabai. This interplay between standard knot theory and minimal surface theory provides significant new techniques to explore outstanding problems. This project explores two particular applications of these techniques. The first is the extension of the standard 1-parameter idea of thin position for a knot to multi-parameter families. These extensions are used to approach some long-standing problems in knot theory. Connections between multi-parameter thin position and other well-known invariants, such as the energy of a knot, another notion of efficient imbedding, are also described. The second application is to a new type of irreducibility for Heegaard splittings that divides splittings into two groups, one of which is atoroidal. The techniques will be used to seek injective immersed surfaces in one group and negatively curved metrics in the other. People originally thought the earth was flat. This was a reasonable hypothesis, arrived at by considering the local information available at the time. Similarly, by considering the local information available to us, we might hypothesize that the spatial universe we live in is 3-dimensional Euclidean space. Evidence from physics makes th is unlikely. The question then is, what is the 3-dimensional shape of the universe we live in? Is it the 3-dimensional analog of the sphere? Perhaps -- but here we encounter a deep mathematical problem. Unlike 2-dimensional spaces, which are well-understood, there is no list of all 3-dimensional spaces. So we don't even understand what the possibilities are for the shape of the 3-dimensional universe. Perhaps the most important problem in low-dimensional topology today is to find such a list. This project uses the interplay between new minimization techniques from piecewise linear geometry and knot theory to explore aspects of this problem. ***

小行星9704140 三维拓扑学中的一个基本问题是三维流形的分类问题，即找到一个完整的三维流形列表，其中每个空间只出现一次。 二维流形的分类提供了一个三维流形分类的模型。 分类问题的关键一步是由J. H. Rubinstein在1992年描述了一个算法，用来判断一个给定的三维流形是否同胚于三维球面，即“最简单的”三维流形。 他的论点使用最小曲面理论的背景下，三角形3流形。 在这种极小曲面理论和三维球面中结点的薄位置的概念之间有着深刻的联系，这是D。加拜 标准纽结理论和最小曲面理论之间的相互作用为探索突出问题提供了重要的新技术。 这个项目探讨了这些技术的两个特殊应用。 第一个是扩展的标准1参数的想法薄的位置为一个结多参数的家庭。 这些扩展被用来解决纽结理论中一些长期存在的问题。 多参数薄的位置和其他著名的不变量之间的连接，如一个结的能量，另一个概念的有效嵌入，也被描述。 第二个应用是Heegaard分裂的一种新的不可约化，它把分裂分成两组，其中一组是超环面的。 该技术将被用来寻求内射浸没表面在一组和负弯曲度量在其他。 人们最初认为地球是平的。 这是一个合理的假设，是根据当时当地的信息得出的。 同样，通过考虑我们可获得的局部信息，我们可以假设我们生活的空间宇宙是三维欧几里得空间。 物理学的证据表明这是不可能的。 问题是，我们生活的宇宙的三维形状是什么？ 它是球体的三维模拟吗？ 也许吧，但我们在这里遇到了一个深刻的数学问题。 不像二维空间，这是很好理解的，没有所有的三维空间的列表。 所以我们甚至不知道三维宇宙形状的可能性。 也许在低维拓扑学中最重要的问题就是找到这样一个列表。 这个项目使用新的最小化技术之间的相互作用，从分段线性几何和纽结理论，探索这个问题的各个方面。 ***