Mathematical Sciences: Embeddings and Immersions in S3

这个项目的主题是了解某些类型的 的表面和二维物体可以嵌入，沉浸， 并且通常位于三维球体中。 三 正在进行的合作项目和一个单独的项目， 关于（1）封闭辫子代表的链接，与琼S。 （2）平面和布线 consumption猜想，with Abigail阿比盖尔A.汤普森大学 加州，戴维斯，（3）嵌入和沉浸在高度 与Morwen B交替链接外部。英国的Thistlethwaite 田纳西大学，（4）Milnor-Bennequin猜想 和解开的号码。 (1)寻找更好的计算方法 这种联系，可能导致更有效的 算法比哈肯的区分定向链接。 （三） 建立在最近的一次联合胜利上， Tait flyping猜想 纽结和它们的泛化，链接，是相当初级的 几何物体的有趣特性是 拓扑的 我们的意思是，两个几何结不 如果其中一个可以转化为 看起来就像另一个没有削减或解开它，只是通过 推动它的弦来重新排列交叉点。 Topologists 说它们是同一个的两种不同的几何实现， 拓扑纽结 然而，这不是一件小事， 当一个复杂的几何纽结在拓扑上 不同于另一个，而不仅仅是不同的几何形状， 实现。 这个问题可以通过计算某些 数或多项式被称为“拓扑不变量”， 这意味着它们对于不同的 相同拓扑结的几何实现。 问题 如果有一个不变量， 也总是有不同的价值观的几何实现， 不同的拓扑结，但生活并不那么简单--没有单一的 不变量实现了这一理想，甚至所有已知的不变量也没有。 合在一起。 因此，研究新的 不变量，其中一些最有用的是那些启发在最近 量子物理学的想法。 调查人员已经在 这项活动的厚度，并已特别成功， 应用新的多项式不变量解决老问题 关于交替打结的 他将继续利用这些 技术以及一些几何学的。

The theme of this project is understanding how certain types of surfaces and 2-dimensional objects can be embedded, immersed, and, in general, positioned in the 3-dimensional sphere. Three ongoing collaborative projects and one individual one are involved, concerning (1) links represented by closed braids, with Joan S. Birman of Columbia University, (2) planar surfaces and the cabling conjecture, with Abigail A. Thompson of the University of California, Davis, (3) embeddings and immersions in highly alternating link exteriors, with Morwen B. Thistlethwaite of the University of Tennessee, and (4) the Milnor-Bennequin conjecture and unknotting number. (1) seeks better ways of calculating about such links, possibly leading to a more efficient effective algorithm than Haken's for distinguishing oriented links. (3) builds upon a recent joint triumph in settling the hundred-year-old Tait flyping conjecture. Knots and their generalization, links, 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 as well as some more geometric ones.