Knots and 3-Manifolds

摘要奖：DMS-0104126首席研究员：阿比盖尔·A·汤普森拟议的研究重点是结点和三维空间的研究。我们将研究关于纽结的外科手术，三维空间中图的纽结，以及三维流形的Heegaard分解。我们描述了三个具体的目标及其与该领域一些基本问题的联系。第一个目标是发展多参数薄层定位和扫掠的工具和技术，以便它们可以用来解决纽结理论和三维流形理论中的一些长期存在的问题。其次，将作者和M.Scharlemann关于3-球面上未结平面图的工作推广到更一般的情形，最后，通过对Heegaard亏格2的3-流形的更深入的理解，对3-流形的一般问题有了更清晰的理解。我们将研究关于隧道第一个结的具体问题，更大的问题的特殊情况，例如通过对3-球面中的一个结进行手术可以获得哪种属两流形，这些流形中的哪些流形包含具有内射基本群的浸没曲面，以及这些流形的Heegaard图中的信息如何转化为关于流形本身的几何和代数信息。我们生活在一个明显具有三维空间维度的宇宙中。探索我们的宇宙可能存在的可能形式，不仅是物理学家面临的一个深刻问题，也是数学家面临的一个深层次问题。不同的数学可能性会导致非常不同的预期物理性质。例如，我们不知道，如果我们能够将一枚非常快的火箭“直接”送入太空，它最终是否会像地球表面的一艘船一样返回起点，或者它是否会永远离开我们。宇宙的形状有无限的可能性。这个项目建议使用低维拓扑和纽结理论的技术来探索其中的一些可能性。

AbstractAward: DMS-0104126Principal Investigator: Abigail A. ThompsonThe focus of the proposed research is the study of knots and3-dimensional spaces. We will examine questions about surgery onknots, knotting of graphs in 3-space dimensional spaces, andHeegaard decompositions of 3-manifolds. We describe threespecific goals and their connections to some of the basicquestions in the field. The first goal is to develop the toolsand techniques of multi-parameter thin position and sweep-outs sothat they can be used to tackle some of the long-standingproblems in knot theory and 3-manifold theory. A second is toextend work of the proposer and M. Scharlemann on unknottedplanar graphs in the 3-sphere to a more general setting.Finally, we aim to obtain a clearer understanding of generalquestions about 3-manifolds by obtaining a deeper understandingof 3-manifolds of Heegaard genus two. We will examine specificquestions about tunnel number one knots, special cases of largerquestions about what genus two manifolds can be obtained bysurgery on a knot in the 3-sphere, which of these manifoldscontain immersed surfaces with injective fundamental groups, andhow information from the Heegaard diagrams of these manifoldstranslates into geometric and algebraic information about themanifold itself.We live in what is apparently a universe with three spatialdimensions. Exploring the possible forms that our universe mighthave is not only a deep problem for physicists but also formathematicians. Different mathematical possibilities lead tovery different expected physical properties. As an example, wedo do not know if, were we able to program a very fast rocket togo "straight" into space, it would eventually return to itsstarting point, like a ship on the surface of the earth, or if itwould continue to travel away from us forever. There are aninfinite number of possibilities for the shape of the universe.This project proposes to explore some of them using thetechniques of low-dimensional topology and knot theory.