Mathematical Sciences: Knot Invariants and Representation Varieties

点击翻译按钮获取中文摘要

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.)