Mathematical Sciences: Knots, 3-Manifolds and Thin Position

三维拓扑学中的一个基本问题是3-流形的分类问题，即找到一个3-流形的完整列表，其中每个空间只出现一次。二维流形的分类为三维流形的分类提供了一种模型。j·h·鲁宾斯坦在1992年完成了分类问题的关键一步，当时他描述了一种算法来确定给定的3-流形是否同胚于3-球，即“最简单”的3-流形。他的论证在三角化3流形的背景下使用了最小曲面理论。这种最小曲面理论与D. Gabai提出的3球中结点的薄位置概念之间存在着深刻的联系。标准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。本项目探讨了这些技术的两个特定应用。首先是将标准的单参数结点薄位置思想推广到多参数族。这些扩展用于解决结理论中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，例如结的能量，另一个有效嵌入的概念。第二个应用是heegard分裂的一种新型不可约性，它将分裂分为两组，其中一组是单向分裂。该技术将用于在一组中寻找注入浸入表面，在另一组中寻找负弯曲指标。人们最初认为地球是平的。这是一个合理的假设，是通过考虑当时当地可用的信息得出的。同样地，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。物理学的证据表明这是不可能的。那么问题来了，我们生活的宇宙的三维形状是怎样的？它是球体的三维模拟物吗？也许吧——但这里我们遇到了一个深刻的数学问题。不像二维空间，这是很容易理解的，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是什么。也许当今低维拓扑中最重要的问题就是找到这样一个列表。本项目利用分段线性几何的新最小化技术和结理论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. ***