Mathematical Sciences: Knot Theory: New Invariants and TheirTopology

林将研究各种新的结不变量和相关的拓扑。在这个项目的第一部分，他打算把结签名放到3-流形的瞬时不变量的框架中。在等变Floer理论的背景下考虑，该签名不变量有望为结理论中的一些老问题提供新的见解，例如解结数的估计。在第二部分中，他将探讨结多项式与瓦西里耶夫结不变量之间建立的关系。他期望从瓦西里耶夫的初始数据中找到结不变量的组合结构。这些结不变量基本上决定了所有由量子群导出的广义琼斯多项式。他还希望从Vassiliev的结不变量中得到一些关于用平凡琼斯多项式寻找非平凡结的突出问题的信息。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。特别是，结理论在DNA长链生物学上的应用已经借鉴了这些新的不变量的知识。

Lin will study various new knot invariants and related topology. In the first part of this project, he intends to put the knot signature into the framework of instanton invariants of 3- manifolds. Considered in the context of an equivariant Floer theory, the signature invariant is expected to provide a new insight into some old problems in knot theory, e.g., the estimate of the unknotting number. In the second part, he will explore the relationship established between knot polynomials and Vassiliev's knot invariants. He anticipates finding a combinatorial construction of knot invariants from Vassiliev's initial data. These knot invariants determine essentially all generalized Jones polynomials derived via quantum groups. He also hopes to get some information about the outstanding problem of finding non-trivial knots with trivial Jones polynomials from Vassiliev's knot invariants. 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. In particular, applications of knot theory to the biology of long strands of DNA have drawn upon knowledge of these newer invariants.