Topology and Sweep-Out Combinatorics Near Dimension Three

提出者最近的研究集中在纽结理论和三维流形理论，特别是三维流形的Heegaard分裂和相关的unknotting隧道概念。 主要的方法可能被称为新古典的，因为它侧重于包含在三维流形（一个经典的方法），但在一个更复杂的方式曲面的行为。 例如，在表面相交的组合论证中，图论论证线在Gabai的缝合三维流形理论中，与参数化曲面的相交是将信息从层次的末尾传递回开始的主要机制;极小极大原理通常被称为“稀疏化”，它利用静止曲面和运动曲面之间的交集来理解整个流形的拓扑结构。比较两组运动曲面之间的交集（使用瑟夫理论的工具）可以更有效，例如，戈登-纽结补猜想的Luecke解或提议者与Rubinstein关于Heegaard稳定化的显式边界的工作。兴趣，但也因为希望是技术开发的解决方案将指出的方式，以解决更大的问题，更普遍的是，让洞察到一般结构的打结现象（广义解释）和结构的3-流形，特别是结构的边缘和相互作用的自然问题约4-流形。 这个提议的一个基本主题是，通过一种比微分几何更灵活但比简单组合学更结构化的几何方法，关于纽结理论和三维流形理论还有很多东西要学。 这一中间基础的特点是使用了薄位置论证，在这种论证中，通过水平面对三维空间的扫出或通过Heegaard曲面对三维流形的扫出可以揭示出用其他技术可能难以发现的结构。对我们周围世界的最基本观察之一，显然是从我们出生开始，就是它是三维的。 因此，理解具有这种特性的物体是很有趣的：任何生活在物体中的人都将把他们的世界看作是三维的。 这样的物体被称为三维流形，这项研究计划的主要目标是增加我们对它们的理解。 Three-manifold流形supportinteresting有趣phenomena现象. 其中一种现象是打结，一个简单的物体，如一根花园软管（或一串DNA），可以被操纵，使其在空间中的定位相当复杂。 更一般地说，像化学分子这样的物体可以以非常复杂的方式放在一个3-流形中，如果你把它们的部件想象成可以打结和交织的橡胶。 正在开发的理解三维流形的工具帮助我们理解打结，反过来，理解打结也帮助我们理解三维流形。这个正在进行的研究计划的一个成功之处是提出者（与Abigail Thompson）解决了图平面性问题：有一个简单的标准，算法执行，它决定了三维空间中的打结图（e. G.一种化学分子）可以被同位素排列在一个平面上，因此实际上是不打结的。

The proposer's recent research has centered on knot theory and thetheory of 3-manifolds, particularly Heegaard splittings of 3-manifoldsand related notion of unknotting tunnels. The main approach mighthumorously be called neo-classical, for it focuses on the behavior ofsurfaces contained in the 3-manifolds (a classical approach) but in amore sophisticated way. For example, in combinatorial arguments onsurface intersections, graph-theoretic lines of argument (on the arcsof intersection) have proven to be almost magically effective indistilling geometric information; in Gabai's theory of sutured3-manifolds, intersection with a `parameterizing surface is the chiefmechanism for transmitting information from the end of the hierarchyback to the beginning; the minimax principle usually called ``thinposition" exploits the intersection between a stationary surface and amoving surface to understand the topology of the entire manfold.Comparing intersections between two sets of moving surfaces (using thetools of Cerf theory) can be even more effective as in, for example,the Gordon-Luecke solution to the knot complement conjecture or theproposer's work with Rubinstein on explicit bounds for Heegaardstabilization.The problems addressed in this proposal have been chosen not solelyfor their intrinsic interest but also because the hope is thattechniques developed in their solution would point the way to thesolution of grander problems and, more generally, give insight to thegeneral structure of knotting phenomena (broadly construed) and thestructure of 3-manifolds, particularly structure that is on the edgeof and interacts with natural questions about 4-manifolds. Anunderlying theme of this proposal is that there is much to be learnedabout knot theory and the theory of 3-manifolds via a geometricapproach that is more flexible than differential geometry but morestructured than simple combinatorics. This medium ground ischaracterized by a use of thin position arguments, in which sweep-outsof 3-space by level planes or sweep-outs of 3-manifolds by Heegaardsurfaces can reveal structure that might be difficult to find by othertechniques.One of the most basic observations about the world around us, apparentalmost from our birth, is that it is 3-dimensional. So it is ofinterest to understand objects with this property: anyone living inthe object would see their world as 3-dimensional. Such objects arecalled 3-manifolds, and the broad goal of this research proposal is toincrease our understanding of them. Three-manifolds supportinteresting phenomena. One of these phenomena is knotting, in which asimple object like a garden-hose (or a string of DNA) can bemaneuvered so that its positioning in space is quite complex. Moregenerally, objects like chemical molecules can be put in a 3-manifoldin extraordinarily complex ways if one thinks of their parts as madeof rubber which can be knotted and interweaved. Tools which are beingdeveloped to understand 3-manifolds help us understand knotting and,conversely, understanding knotting helps us understand 3-manifolds.One success of this ongoing research program is the proposer'ssolution (with Abigail Thompson) of the graph-planarity problem: Thereis a simple criterion, algorithmic in execution, which determineswhether a knotted graph in 3-space (e. g. a chemical molecule) can beisotoped to lie in a plane, and so in fact is unknotted.