Problems in Low Dimensional Topology

DMS-0306062 William Menasco研究人员的研究集中在理解低维拓扑中的两个重要现象： 第一，准确地理解通过使用稳定化来关联两个等价的闭合辫子所实现的;第二，理解何时在三维流形内出现拓扑本质表面，即何时是三维流形哈肯。 经典的稳定化结果是“马尔可夫定理”，它说的是，任何两个封闭的辫子代表相同的定向链接类型的3球是相互相关的，通过一系列的移动（同位素）：共轭，稳定和不稳定。不幸的是，马尔可夫定理只说明了一个序列的存在，但直到最近，对这个稳定化序列的确切理解在很大程度上仍然是一个大黑盒子。第一个进入稳定化黑盒子的是“无稳定化马尔可夫定理”（MTWS），这是我和琼·伯曼长期合作的成果。 MTWS可以准确地告诉人们在编织物之间的同位素中实现了什么稳定，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学领域利用这种理解。三维流形内的本质曲面告诉我们空间的几何形状。 不是所有的3-流形都包含本质曲面，但是对于给定的3-流形M， 缺少本质曲面，存在另一个具有本质曲面的3-流形M'，且M'“覆盖”M。 理解三维流形M何时有这样一个对应的覆盖M'是“虚哈肯猜想”的核心问题。 与约瑟夫·马斯特斯合作的调查员& 张兴儒正在寻求一种新的策略来在一般环境下攻击这一猜想。我们所生活的三维空间是独一无二的，它能够保留关于闭环集合（想想珠宝盒里一串纠缠在一起的珍珠）中“最打结”的信息。 这种打结现象不会发生在任何更低或更高维度的空间-一个5维珠宝盒的好处是，一串珍珠永远不会缠结在一起。 因此，正如任何研究DNA链的微生物学家会告诉你的那样，打结是我们三维存在的一个重要特征，需要加以理解。 基本问题出现了。什么时候不同的两个结（单股）或链接（多股）说明相同类型的knottest-也就是说，什么时候有一个链接的一系列运动，使其在三维空间中移动，使其看起来像另一个链接？ 什么时候一个看起来纠结的结，实际上通过运动等价于一个可以平放在平面上的简单圆？运动可以是非常复杂的（想想试图解开一团鱼线）。 研究人员的研究集中在理解和编纂这些运动（与琼伯曼合作，“马尔可夫定理没有稳定”）。 同样，在试图理解三维空间时，我们可以研究出现在其中的“基本”表面，即在空间中不能被碰撞成一点的表面。 这样的表面可以给我们提供一种在太空中导航的坐标系统-天文学家对确定我们的宇宙是否有任何基本的表面非常感兴趣。 如果一个三维空间有一个本质面，那么它被称为“哈肯”。 并非所有的空间都是哈肯的，但有些不是哈肯的空间可以被哈肯的空间“涂上”或“覆盖”，也就是说，它们可能是“虚拟哈肯”。& Xingru Zhang正在追求一种新的策略，确定何时三维空间是虚拟的。

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.