Knots and 3-Manifolds

项目编号：dms -0104126项目负责人：Abigail A. thompson，主要研究方向为结和三维空间。我们将研究关于结的手术，三维空间图的结，以及三维流形的严格分解的问题。我们描述了三个具体目标以及它们与该领域一些基本问题的联系。第一个目标是开发多参数薄定位和扫描的工具和技术，以便它们可以用于解决结理论和三流形理论中一些长期存在的问题。第二种是将提出者和M. Scharlemann关于3球无结平面图的工作推广到更一般的情况。最后，我们的目标是通过对Heegaard属2的3-流形的更深入的理解来获得对关于3-流形的一般问题的更清晰的理解。我们将研究关于隧道1号结的具体问题，更大问题的特殊情况，关于在3球中的一个结上进行手术可以得到什么属二流形，这些流形中哪些包含具有注入基本群的浸入曲面，以及这些流形的heegard图中的信息如何转化为流形本身的几何和代数信息。我们生活在一个三维空间的宇宙中。探索我们的宇宙可能存在的形式不仅是物理学家的一个深刻问题，也是数学家的一个深刻问题。不同的数学可能性导致不同的预期物理性质。例如，我们不知道，如果我们能够编程一个非常快的火箭“直接”进入太空，它最终会回到它的起点，就像地球表面的一艘船，或者它会继续永远远离我们。宇宙的形状有无数种可能。本项目拟利用低维拓扑和结理论的技术来探索其中的一些问题。

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.