Geometric Approach To Braid Theory

本计画的重点是应用几何技巧来研究辫状理论，并试图证明纽结理论的新结果。特别是，使用在她最近的工作中开发的方法，主要研究者将研究辫子指数，经典的结不变量，和各种几何量，如扭曲，HOMFLY多项式，自链接数和Khovanov-Rozansky同源性之间的关系。后者提供了一个特别新的和强有力的工具，研究问题的辫子理论。PI还打算探索使用开卷分解，接触和Legendrian手术来分析negative-flype-braid-move的可能性，这对于接触3-流形中的横向结的分类起着至关重要的作用。一个目标是找到一个新的横截纽结不变量sensible负flype-moves.Braiding出现在代数几何，密码学，动力系统，同伦群的领域，算子代数，和机器人。此外，结和链接理论广泛适用于各种科学学科，包括生物学和医学，例如，蛋白质折叠机制和DNA结构起着开创性的作用。该项目的影响可能会刺激低维拓扑学与数学，理论物理和生物学等领域之间的交流和合作。

The focus of this project is application of geometric techniques to study braid theory with the intention to prove new results in knots theory. In particular, using methods developed in her recent work, the principal investigator will study the relationship between the braid index, a classical knot invariant, and various geometric quantities such as writhes, HOMFLY polynomials, self-linking number, and Khovanov-Rozansky homology. The latter provides an especially new and potent tool to study problems in braid theory. The PI also intends to explore the possibility of using open book decomposition, contact and Legendrian surgeries to analyze negative-flype-braid- moves, which have been playing a crucial role for classification of transversal knots in contact 3-manifolds. A goal is to find a new transversal knot invariant sensible to negative-flype-moves.Braiding appears unexpectedly in algebraic geometry, cryptography, dynamical systems, homotopy groups of spheres, operator algebras, and robotics. In addition, knot and link theory is broadly applicable in a variety of scientific disciplines including biology and medicine where, for example, protein folding mechanisms and DNA structure play a seminal role. The impact of the project will likely stimulate communication and collaboration between low-dimensional topology and other fields of mathematics, theoretical physics and biology.