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

这位研究人员和纽约州立大学布法罗分校的威廉·W·梅纳斯科(William W.Menasco)最近证明了古老的交错结泰特猜想和交错结缆线猜想的增强版本。他们将扩大对与特定链接族有关的几何问题的研究。他们将寻求在特定的类别中描述解开第一类的结的特征，例如交替结、孟德斯诺结和树状结的类别。他们希望建立一种有效的程序来确定这些类别中的解结数量。他们还将寻求有关电缆纽结的解开数的信息，并将研究关于连通和解结数是可加的这一古老猜想。他们将试图证明交错纽结具有性质P，交错纽结的所有边界斜率都是偶数，并且交错环由它们的补集决定。他们将努力开发在交错结外部构造基本叠层的技术，并且在边界斜率为有理的情况下，他们将检查通过适当的Dehn填充获得的闭合流形中这些叠层的完成度。结是相当基本的几何对象，其真正有趣的属性是拓扑。我们的意思是，如果两个几何纽结中的一个可以在不切断或解开它的情况下被变换成与另一个一样的样子，那么两个几何纽结就没有什么有趣的区别，只需推动它的线来重新排列交叉点。拓扑学家说，它们是同一个拓扑结的两个不同的几何实现。然而，当一个复杂的几何节点在拓扑上与另一个不同时，认识到这不是一件微不足道的事情，而不仅仅是不同的几何实现。这个问题可以通过计算某些数字或多项式来解决，这些数字或多项式被称为“拓扑不变量”，这意味着对于同一拓扑结点的不同几何实现，它们总是具有相同的值。如果有一个不变量对不同的拓扑节点的几何实现总是有不同的值，那么这个问题就会简化为纯代数，但生活并不那么简单--没有一个单一的不变量达到这个理想，甚至所有已知的不变量加在一起也不是。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思想启发的那些不变量。这位研究人员一直处于这一活动的中心，并特别成功地应用了一些新的多项式不变量来解决关于交替纽结的旧问题。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.