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

这位研究者和纽约州立大学布法罗分校的威廉·w·梅纳斯科（William W. Menasco）最近证明了令人尊敬的泰特交替结猜想的改进版本，以及交替结的布线猜想。他们将把研究范围扩大到有关特定链族的几何问题。他们将试图在某些类别中描述解结第一的结，例如交替结、蒙特西诺斯结和树状结。他们希望建立一个有效的程序来确定这些类别中结的解开数量。他们还将寻求关于绳结的解结数的信息，并将研究解结数相对于连通和是可加性的旧猜想。他们将试图证明交替结具有性质P，所有交替结的边界斜率都是偶数，并且交替连杆由它们的补决定。他们将努力开发在交替结外部构造基本层合的技术，在边界斜率为合理的情况下，他们将在通过适当的Dehn填充获得的封闭流形中检查这些层合的完成情况。结是相当基本的几何对象，其真正有趣的性质是拓扑。我们的意思是，两个几何结不会以一种有趣的方式有所不同，如果其中一个可以在不切割或解开它的情况下转换成与另一个相同的样子，只需要推动它的绳子重新排列交叉点。拓扑学家说它们是同一拓扑结的两种不同的几何实现。然而，当一个复杂的几何结在拓扑结构上与另一个不同时，识别它不是一件小事，而不仅仅是不同的几何实现。这个问题可以通过计算某些被称为“拓扑不变量”的数字或多项式来解决，这意味着对于相同拓扑结的不同几何实现，它们总是具有相同的值。如果有一个不变量，对于不同拓扑结点的几何实现总是有不同的值，那么这个问题将被简化为纯代数，但生活并没有那么简单——没有一个不变量能达到这种理想，甚至所有已知的不变量都不能。因此，研究新的不变量是有价值的，其中一些最有用的是近年来受到量子物理学思想启发的不变量。研究人员一直在从事这项工作，并特别成功地应用了一些新的多项式不变量来解决有关交替结的老问题。他将继续利用这些技术。

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.