Mathematical Sciences: Topological Knot Theoretic Connections

这项研究将集中在纽结理论中的不变量琼斯多项式上，并将使用一种新的拓扑方法，用一种更简单的几何方法来取代代数和组合技术。这项研究应该通过用其他相关知识领域的语言来调查一个领域提出的问题，从而加深对该领域问题的理解。研究这一新的几何方法与已确立的几何方法之间的关系，应该有助于更好地理解所有这一理论，并提供一种统一的方法。纽结理论是拓扑学和代数中的一个研究领域，它在描述DNA的数学生物学和理论物理中有各种应用；特别是保角场论和统计力学。

This research will focus on the Jones polynomial, an invariant in knot theory, and will use a new topological approach which replaces the algebraic and combinatorial techniques with a simpler geometric method. This research should give a deeper understanding of a problem posed in one field by investigating it in the language of other areas of connected knowledge. The study of the relations between this new geometric approach and those already established should lead to a better understanding of all of this theory and provide a unified approach. Knot theory is an area of research in topology and algebra that has a variety of applications to mathematical biology in the description of DNA and to theoretical physics; in particular, conformal field theory and statistical mechanics.