Mathematical Sciences: Polynomial Invariants in Knot Theory

本研究集中于结理论中的多项式不变量，它们的结构和拓扑意义，以及它们与数学物理的相互关系。特别是，该项目专注于研究结多项式的状态模型。状态模型首先是由Alexander-Conway多项式和Jones多项式的首席研究员引入的。该项目研究了这些模型和相关模型（Yang-Baxter模型），涉及统计力学和量子群，以及它们与更一般的skein多项式状态模型（Homfly和Kauffman）的关系，这些模型本质上是组合的。主要研究者打算使用状态模型来探索拓扑问题，并探索它们与这些不变量的可能的内在（无图）定义的联系。关于一元和二元多项式的一些相当简单的代数已经被应用于区分本质上不同的结的问题（而不是仅仅是同一结的不同表示）。虽然这个想法很简单，但重要的应用存在于遗传物质DNA的盘绕和可能的打结，以及统计力学中的问题。

This research centers on polynomial invariants in knot theory, their structure and implications for topology, and their interrelationships with mathematical physics. In particular, the project concentrates on the investigation of state models for knot polynomials. State models were first introduced by the principal investigator for the Alexander-Conway polynomial, and for the Jones polynomial. The project investigates these models and related models (Yang-Baxter models) involving statistical mechanics and quantum groups, and their relations with more general state models for the skein polynomial (Homfly and Kauffman) that are combinatorial in nature. The principal investigator intends to use state models to explore topological problems and to explore their connection with possible intrinsic (diagram-free) definitions of these invariants. Some fairly simple algebra concerning polynomials in one and two variables has found application to the problem of distinguishing between essentially different knots (as opposed to merely different presentations of the same knot). Simple though the idea is, important applications exist to the coiling and possible knotting of the genetic material DNA, as well as to problems in statistical mechanics.