Mathematical Sciences: Polynomial Invariants in Knot Theory

本文主要研究纽结中的多项式不变量 理论，它们的结构和对拓扑学的影响，以及 它们与数学物理的相互关系。 在 特别是，该项目集中于调查 纽结多项式的状态模型 国家模型首先是 由首席研究员介绍， Alexander-Conway多项式和Jones多项式。 的 项目研究这些模型和相关模型 （杨-巴克斯特模型）涉及统计力学和 量子群及其与更一般态的关系 模型的绞股多项式（Homfly和考夫曼）， 组合的。 主要研究者打算 使用状态模型来探索拓扑问题， 它们与可能的内在联系（无干扰） 这些不变量的定义。 一些相当简单的代数涉及多项式在一个 和两个变量的问题 区分本质上不同的结（相对于 仅仅是同一个结的不同表现）。 简单 虽然这个想法是，重要的应用存在的盘绕 以及遗传物质DNA的可能打结， 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.