Geometric Approach To Braid Theory

本项目的重点是应用几何技术研究编织理论，旨在证明纽结理论的新结果。特别是，使用她最近工作中开发的方法，主要研究人员将研究辫子指数，一个经典的纽结不变量，和各种几何量之间的关系，如扭度，HOMFLY多项式，自链接数，和Khovanov-Rozansky同调。后者为研究辫子理论中的问题提供了一个特别新而有力的工具。PI还打算探索使用开卷分解、接触和Legendrian手术来分析负Flype-Braid运动的可能性，这些运动在Contact 3-流形中的横向结的分类中一直发挥着关键作用。一个新的目标是找到一种新的对负平移敏感的横截纽结。编织出人意料地出现在代数几何、密码学、动力系统、球面同伦群、算子代数和机器人学中。此外，结链理论广泛适用于包括生物学和医学在内的各种科学学科，例如，蛋白质折叠机制和DNA结构在这些学科中发挥着开创性的作用。该项目的影响可能会促进低维拓扑学与其他数学、理论物理和生物学领域之间的交流与合作。

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.