Geometric Approach To Braid Theory

本课题的重点是将几何技术应用于编织理论的研究，以证明结理论的新结果。特别是，使用她最近工作中开发的方法，首席研究员将研究辫指数（经典结不变量）与各种几何量（如wrthes， HOMFLY多项式，自连接数和Khovanov-Rozansky同调）之间的关系。后者为研究辫状结构理论中的问题提供了一种全新而有力的工具。PI还打算探索使用开卷分解，接触和Legendrian手术来分析负飞-辫-移动的可能性，这在接触3流形的横向结分类中起着至关重要的作用。我们的目标是找到一个新的对负飞动敏感的横向结不变量。编织出人意料地出现在代数几何、密码学、动力系统、球的同伦群、算子代数和机器人中。此外，结和链接理论广泛应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重要作用。该项目的影响可能会刺激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.