Problems in Low Dimensional Topology

研究人员的研究集中于理解低维拓扑中的两个重要现象：第一，通过使用稳定化将两个等价的封闭辫子联系起来，准确地理解了什么；第二，理解在三维流形内何时出现拓扑本质曲面，即何时是三维流形。经典的稳定化结果是“马尔可夫定理”，该定理指出，三维球面上同一定向连杆类型的任意两个闭合辫子代表通过一系列运动(同位素)相互联系：共轭、稳定和失稳。不幸的是，马尔可夫定理只说序列存在，但直到最近，对这个稳定序列实现的确切理解在很大程度上仍然是一个大黑盒。第一次窥探稳定黑盒的是没有稳定的马尔可夫定理(MTWS)，这是与琼·比曼长期合作的产物。MTW可以准确地告诉人们在辫子之间的等价性中实现了什么稳定，该项目的主要目标之一是在链接分类、链接不变量和接触几何领域利用这种理解。3-流形内的一个基本曲面告诉我们空间的几何。并不是所有的三维流形都包含本质曲面，但对于一个给定的缺乏本质曲面的三维流形M，有可能存在另一个具有本质曲面的三维流形M‘和M’“覆盖”M。理解当一个三维流形M有这样一个对应的覆盖M‘是“虚哈肯猜想”的焦点。这位研究人员与约瑟夫·马斯特斯和张兴如合作，正在寻求一种新的策略，在一般情况下破解这一猜想。我们生活的三维空间的独特之处在于，它能够保存一系列闭合环(想想珠宝盒中一串纠结的珍珠)的“最结点”信息。这种打结测试的现象不会出现在任何更低或更高维的空间-5维珠宝盒的优点是一串珍珠永远不会纠缠在一起。因此，正如任何研究DNA链的微生物学家都会告诉你的那样，结测试是我们需要理解的三维存在的一个重要特征。基本的问题就出现了。什么时候不同的两个结(单股)或链接(多股)说明同一类型的结测试-也就是，什么时候一个链接的一系列运动在三维空间中移动它，使它看起来像另一个链接？什么时候是一个看起来纠缠在一起的结，通过运动实际上是等价的，一个可以平放在平面上的简单圆圈？运动可能非常复杂(想想试图解开一大堆钓鱼线)。这位研究人员的研究重点是理解和编码这些运动(与琼·比尔曼合作出版的《没有稳定的马尔可夫定理》)。在试图理解3维空间的过程中，我们还可以研究出现在其中的“基本”表面，即空间中不能被碰撞到一点的表面。这样的表面可以为我们提供一种在太空中导航的坐标系-天文学家对确定我们的宇宙是否有任何必要的表面非常感兴趣。如果一个三维空间有一个本质表面，那么它被称为“哈肯”。并不是所有的空间都是Haken的，但有些不是Haken的空间可以被Haken的空间覆盖，也就是说，它们可能是“虚拟的Haken”。研究人员与约瑟夫·马斯特斯和张兴如合作，正在探索一种新的策略，以确定三维空间何时是虚拟的Haken。

DMS-0306062William MenascoThe investigator's research is focussed on understanding twoimportant phenomena in low dimensional topology: first, understanding exactly what is accomplished through the use of stabilization in relating two equivalent closed braids; and second, understanding when there is the occurence of topologically essential surfaces inside a 3-dimensional manifold, i.e. when is a 3-manifold Haken. The classical stabilization result is "Markov's Theorem" which says that any two closed braid representatives of the same oriented link type in the 3-sphere are related to each other through a sequence of moves (isotopies): conjugation, stabilization and destabilization. Unforunately, the Markov Theorem only saysa sequence exists, but understanding exactly what this stabilization sequenceaccomplishes has largely remained a big black box until recently.The first peek inside the stabilization black box is the "Markov Theorem Without Stabilization" (MTWS), a product of a long collaborative effort with Joan Birman. The MTWS can tell one exactly what stabilization achieves in an isotopy between braids and one of the main goals of the project is to exploit this understanding in the areas of link classification, link invariants and contact geometry. An essential surface inside a 3-manifold tells one about the geometry of the space. Not all 3-manifolds contain essentialsurfaces, but it is possible that for a given 3-manifold M that is lacking any essential surface there is another 3-manifold M' which hasessential surfaces and M' "covers" M. Understanding when a 3-manifold Mhas a such a corresponding cover M' is the focus of the "Virtual Haken Conjecture". The investigator in collaboration with Joseph Masters & Xingru Zhang is pursuing a new strategy for attacking this conjecture in a general setting.The 3-dimensional space in which we live is unique in its ability to retaininformation about the "knottest" of a collection of closed loops (think ofa collection of tangled strands of pearls inside a jewelry box). This phemomenonof knottest does not occur in any lower or higher dimensional space---theadvantage of a 5-dimensional jewelry box is that a collection ofstrands of pearls can never be tangled. Thus, as any micro-biologist working withDNA strands will tell you, knottest is an important featureof our 3-dimensional existence that needs to be understood. Basic questions arise. When are different two knots (single strands) or links (multiple strands)illustrating the same type of knottest---that is, when is there a sequence ofmotions of one link that move it around in 3-space so that it appears like that other link? When is When is a knot which appears to be tangle in fact equivalent through motions a simple circle that can be laid flat in a plane?Motions can be very complex (think of trying to untanglea mass of fishing line). The investigator's research has focussed on understandingand codifying these motions (in collaboration with Joan Birman, "The Markov TheoremWithout Stabilization"). Also in trying to understand 3-dimensional spaces wecan study "essential" surfaces that occur in them, i.e. surfaces that in the spacecan not be crashed down to a point. Such surfaces can give us a type of coordinate systemfor navigating in the space---astronomers are very interested in determining if ouruniverse has any essential surfaces. If a 3-dimensional space has an essential surfacethen it is called "Haken". Not all spaces are Haken, but some that are not can be'painted over' or "covered" by ones that are, i.e. they may be "Virtually Haken".The investigator in collaboration with Joseph Masters & Xingru Zhang is pursuing a new strategy determining when a 3-dimensional space is Virtually Haken.