Mathematical Sciences: Knot Theory: New Invariants and TheirTopology

林将研究各种新的结不变量和相关的 topology. 在这个项目的第一部分，他打算把 结签名到框架的瞬子不变量的3- 流形 在等变Floer的上下文中考虑 理论，签名不变量预计将提供一个新的 对纽结理论中的一些老问题的认识，例如，的估计 解开的号码 在第二部分中，他将探索 纽结多项式和瓦西里耶夫多项式之间的关系 结不变量 他希望能找到一个组合的 从Vassiliev的初始数据构建结不变量。 这些纽结不变量基本上决定了所有广义Jones 通过量子群导出的多项式。 他也希望能得到一些 关于寻找非平凡的突出问题的信息 Vassiliev纽结的平凡Jones多项式纽结 不变量 纽结是相当基本的几何物体， 有趣的性质是拓扑性质。 我们的意思是， 几何结并没有以有趣的方式不同，如果其中一个 可以在不切割的情况下变形成另一个样子， 解开它，只是通过推动它的字符串重新排列， 交叉路口 拓扑学家说它们是两种不同的几何 相同拓扑结的实现。 然而， 当一个复杂的几何结 在拓扑上与另一个不同，而不仅仅是一个 不同的几何实现。 这个问题可以通过以下方式解决： 计算某些数或多项式，这些数或多项式被称为 “拓扑不变量”，这意味着它们总是具有相同的 相同拓扑的不同几何实现的值 结 这个问题将被简化为纯代数，如果有 一个不变量，它也总是有不同的几何值， 实现不同的拓扑结，但生活并非如此 简单-没有一个单一的不变量达到这个理想，甚至所有的 已知的不变量放在一起。 因此， 研究新的不变量，其中一些最有用的是那些 近年来受到量子物理学的启发。在 特别是，纽结理论在生物学中的应用， DNA链已经利用了这些新的不变量。

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.