Mathematical Sciences: Knots, 3-Manifolds and Thin Position

9704140 Thompson三维拓扑学中的一个基本问题是3-流形的分类问题，即找出每个空间恰好出现一次的3-流形的完整列表。二维流形的分类提供了一种为三维流形寻求的分类模型。J·H·鲁宾斯坦在1992年完成了分类问题的关键一步，当时他描述了一个算法来确定给定的3-流形是否与3-球面同胚，3-球面是最简单的3-流形。他的论点在三角三维流形的背景下使用了极小曲面理论。这类极小曲面理论与D·加拜提出的3-球面节点薄位置的概念有很深的联系。标准纽结理论和极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。这个项目探索了这些技术的两个特殊应用。第一种是将标准的单参数节点薄位置思想推广到多参数族。这些推广被用来探讨纽结理论中的一些长期存在的问题。还描述了多参数薄位置与其他众所周知的不变量之间的联系，例如结点的能量，这是另一个有效嵌入的概念。第二个应用是Heegaard分裂的一种新类型的不可约性，它将分裂分成两组，其中一组是阿托洛特分裂。这些技术将被用来在一组中寻找内射浸没曲面，在另一组中寻找负曲线度量。人们最初认为地球是平的。这是一个合理的假设，是通过考虑当时可获得的当地信息得出的。类似地，通过考虑我们可以获得的局部信息，我们可以假设我们生活的空间宇宙是三维欧几里得空间。来自物理学的证据表明这是不可能的。那么问题是，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--但我们在这里遇到了一个深刻的数学问题。与众所周知的2维空间不同，没有列出所有3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是什么。也许今天低维拓扑学中最重要的问题就是找到这样一个列表。这个项目使用分段线性几何和纽结理论中新的最小化技术之间的相互作用来探索这个问题的各个方面。***

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. ***