Mathematical Sciences: Knot Theory: New Invariants and TheirTopology

林将学习各种新的纽结不变量和相关的拓扑。在这个项目的第一部分，他打算将纽结签名放入三维流形的瞬子不变量的框架中。在等变Floer理论的背景下考虑，签名不变量有望为纽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，他将探讨纽结多项式与瓦西里耶夫纽结不变量之间建立的关系。他期待从瓦西里耶夫的初始数据中找到纽结不变量的组合结构。这些纽结不变量本质上决定了由量子群导出的所有广义琼斯多项式。他还希望从瓦西里耶夫的纽结不变量中得到一些关于寻找具有平凡琼斯多项式的非平凡纽结的悬而未决的问题的一些信息。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。特别是，纽结理论在DNA长链生物学中的应用利用了这些较新的不变量的知识。

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.