Mathematical Sciences: Unknotting Numbers, and Essential Surfaces and Laminations in Knot Exteriors

调查员和威廉·W·纽约州立大学布法罗分校的梅纳斯科 最近证明了一个古老泰特的增强版 交替结猜想，以及电缆猜想 用于交替打结。 他们将扩大调查范围， 关于特定链环族的几何问题。 他们 将试图描述解开第一个结的特点， 某些类别，例如交替结的类别， 蒙特西诺斯结和树结。 他们希望建立一个 确定解结数量的有效程序 在这些班级中。 他们亦会搜集有关 解开一个电缆结的号码，并将调查旧的 猜想解结数关于是可加的 连通和 他们将试图证明，交替结 性质P，所有交错纽结的边界斜率是偶数 而这一切，都是由他们的意志所决定的。 互补。 他们将奋进开发技术， 在交替的结外部构造必要的叠层， 在边界斜率合理的情况下， 在闭流形中检验这些叠层的完备性 通过适当的Dehn灌装获得。 纽结是相当基本的几何物体， 有趣的性质是拓扑性质。 我们的意思是， 几何结并没有以有趣的方式不同，如果其中一个 可以在不切割的情况下变形成另一个样子， 解开它，只是通过推动它的字符串重新排列， 交叉路口 拓扑学家说它们是两种不同的几何 相同拓扑结的实现。 然而， 当一个复杂的几何结 在拓扑上与另一个不同，而不仅仅是一个 不同的几何实现。 这个问题可以通过以下方式解决： 计算某些数或多项式，这些数或多项式被称为 “拓扑不变量”，这意味着它们总是具有相同的 相同拓扑的不同几何实现的值 结 这个问题将被简化为纯代数，如果 一个不变量，它也总是有不同的几何值， 实现不同的拓扑结，但生活并非如此 简单-没有一个单一的不变量达到这个理想，甚至所有的 已知的不变量放在一起。 因此， 研究新的不变量，其中一些最有用的是那些 近年来受到量子物理学的启发。 的 调查员一直在这项活动的厚，并已 特别是成功地应用了一些新的多项式不变量 来解决关于交替打结的老问题 他将继续 来利用这些技术。

The investigator and William W. Menasco of SUNY-Buffalo have recently proved an enhanced version of the venerable Tait conjecture for alternating knots, as well as the cabling conjecture for alternating knots. They will broaden their investigations into geometric questions concerning specific families of links. They will seek to characterize knots of unknotting number one within certain classes, for instance the classes of alternating knots, Montesinos knots, and arborescent knots. They hope to establish an effective procedure for determining the unknotting number of knots within these classes. They will also seek information on the unknotting number of a cabled knot and will investigate the old conjecture that unknotting number is additive with respect to connected sum. They will seek to prove that alternating knots have property P, that all boundary slopes of alternating knots are even integers, and that alternating links are determined by their complements. They will endeavor to develop techniques for constructing essential laminations in alternating knot exteriors, and in the case where the boundary slope is rational, they will examine the completions of these laminations in the closed manifold obtained by appropriate Dehn filling. Knots are rather elementary geometric objects whose really interesting properties are topological. By this we mean that two geometric knots do not differ in an interesting way if one of them can be transformed to look just like the other without cutting or untying it, just by pushing its string about to rearrange the crossings. Topologists say that they are two different geometric realizations of the same topological knot. Nevertheless, it is not a trivial matter to recognize when one complicated geometric knot is topologically different from another, rather than just a different geometric realization. This problem can be addressed by computing certain numbers or polynomials which are called "topological invariants," meaning that they always have the same value for different geometric realizations of the same topological knot. The problem would be reduced to pure algebra if there were one invariant which also always had different values for geometric realizations of different topological knots, but life is not so simple -- no single invariant achieves this ideal, nor even all the known invariants taken together. It is therefore valuable to investigate new invariants, some of the most useful being those inspired in recent years by ideas from quantum physics. The investigator has been in the thick of this activity and has been particularly successful in applying some new polynomial invariants to settle old questions about alternating knots. He will continue to exploit these techniques.