Mathematical Sciences: Knot Invariants and Representation Varieties

该研究项目包括两个部分，都试图 研究结（链接）不变量通过表示品种。 的 第一部分是林书豪工作的延续，他在工作中定义了一个 关于子簇交数的纽结不变量 并将这个纽结不变量与经典的纽结不变量联系起来。 到 推广这项工作，可以定义其他的交集数 他感兴趣的是将这些交集与 经典链接不变量 在第二部分，他提出了一种方法， 为S3中的节点构造一个二元多项式， 推广的定义的亚历山大多项式通过 塞弗特形成。 这种结构与 代表品种。 特别是，它提出了一种可能的 琼斯多项式和卡森不变量之间的联系 同源性3-球。 Lin希望这项研究能够 关于一些重要问题的一些见解，例如如何概括 从S3到任意三维流形的琼斯多项式以及如何 根据拓扑结构解释琼斯多项式， 纽结补集 三维空间中的结是自然物体， study. 林描述的多项式减少了以下问题： 区分真正不等价的结和不同的结 同样的结的呈现到关于多项式的问题， 也就是说，对于相当简单的代数问题。 (This过渡 从几何到代数的转变与其他数学 没有明显的联系）。

This research project consists of two parts, both trying to study knot (link) invariants via representation varieties. The first part is a continuation of Lin's work wherein he defined a knot invariant in terms of an intersection number of subvarieties and related this knot invariant to a classical one. To generalize this work, one can define other intersection numbers and he is interested in relating these intersection numbers with classical link invariants. In the second part, he suggests a way of constructing a 2-variable polynomial for a knot in S3, generalizing the definition of the Alexander polynomial via Seifert forms. This construction is intimately related to representation varieties. In particular, it suggests a possible connection between the Jones polynomial and Casson's invariant of homology 3-spheres. Lin's hope is that this research will shed some light on some important questions, such as how to generalize the Jones polynomial from S3 to arbitrary 3-manifolds and how to interpret the Jones polynomial in terms of the topology of the knot complement. Knots in three-dimensional space are natural objects to study. The polynomials Lin describes reduce the problem of distinguishing truly inequivalent knots from different presentations of the same knot to questions about polynomials, i.e. to fairly simple algebraic problems. (This transition from geometry to algebra turns out to be related to other mathematics of no immediately apparent connection.)