Mathematical Sciences: Knot Invariants and Representation Varieties

本研究项目由两部分组成，都试图通过表示变体来研究结(链)不变量。第一部分是林的工作的继续，他用亚簇的交数定义了一个纽结不变量，并将这个纽结不变量与一个经典的纽结不变量联系起来。为了推广这项工作，人们可以定义其他交数，他感兴趣的是将这些交数与经典的链接不变量联系起来。在第二部分中，他提出了一种构造S3中纽结的二元多项式的方法，通过Seifert形式推广了Alexander多项式的定义。这一构式与表现形式密切相关。特别指出了同调三球面上的Jones多项式与Casson不变量之间可能存在的联系。林希望这项研究能对一些重要问题有所启发，例如如何将琼斯多项式从S3推广到任意的3-流形，以及如何根据纽结补的拓扑来解释Jones多项式。三维空间中的节点是自然研究的对象。林所描述的多项式将从同一结点的不同表示中区分真正不等价的结点的问题归结为关于多项式的问题，即相当简单的代数问题。(这种从几何到代数的转变被证明与其他没有直接明显联系的数学有关。)

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.)