Surfaces and Geometry and Topology of Quantum Link Invariants

纽结理论研究三维空间中的打结的闭合回路，认为是空间的连续变换。在数学中，如果不切断和重新粘合形成环的弦就不能相互转换的结被认为是彼此不同的，或者换句话说，是不等价的。区分不等价结的关键工具称为结不变量，这些基本上是通过空间的连续变换保持的结的属性。在这个项目中，PI将研究量子结不变量之间的关系，这是一个相对不太了解的家族，由量子物理学和数学的思想构建。量子拓扑学的中心问题与三维空间的几何拓扑、由量子不变量指定的多项式的数论性质以及定义理论的量子群的代数结构有着广泛的联系。该项目包括本科生和硕士生的研究计划。PI还将参与“编程女孩”俱乐部会议，为当地社区服务。 有色琼斯多项式是一个重要的纽结不变量，它来自量子群的表示理论，是量子拓扑、低维拓扑和双曲几何的核心。 在这个项目中，PI和她的合作者将通过应用经典的三流形拓扑和法向曲面理论的技术，扩展定义有色琼斯多项式的状态和与链路补中适当嵌入的曲面之间的对应关系。第二部分证明了色Khovanov同调的强斜率猜想的一个范畴化版本，它是色Jones多项式的一个范畴化。PI将评估该框架的潜力，以提供关于Khovanov同源性与结Floer同源性之间关系的新观点，这是另一个链接不变式，已被广泛研究，与其他领域有着深刻的联系。该项目将进一步研究霍瓦诺夫同源性的稳定性特性。这些结果将被用来探索由Plamenevskaya定义的横向不变量的接触几何特性。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Knot theory studies knotted, closed loops in a three-dimensional space considered up to continuous transformations of the space. In mathematics, knots that cannot be transformed into one another without cutting and regluing the strings that form the loops are considered to be different from each other, or in other words, inequivalent. Key tools for distinguishing inequivalent knots are called knot invariants and these are essentially the properties of a knot that are preserved by continuous transformations of the space. In this project the PI will study the relationship between quantum knot invariants, a relatively less understood family, constructed from the ideas of quantum physics and mathematics. Central questions in quantum topology are broadly connected to the geometric topology of three-dimensional spaces, number-theoretic properties of polynomials assigned by quantum invariants, and the algebraic structures of quantum groups defining the theory. The project includes research plans for undergraduate and Master's students. The PI will also engage with "Girls Who Code" club meetings in service to the local community. The colored Jones polynomial is an important knot invariant that comes from the representation theory of quantum groups and lies at the heart of quantum topology, low-dimensional topology, and hyperbolic geometry. In this project the PI and her collaborators will expand the correspondence between a state sum defining the colored Jones polynomial and properly embedded surfaces in a link complement, by applying the techniques of classical three-manifold topology and normal surface theory. The next part is to prove a categorified version of the Strong Slope Conjecture for colored Khovanov homology, which is a categorification of the colored Jones polynomial. The PI will evaluate the potential for this framework to provide new perspectives on the relationship of Khovanov homology to knot Floer homology, another link invariant that has been extensively studied with deep connections to other fields. The project will further study the stability properties of Khovanov homology. These results will be used to explore the contact-geometric properties of the transverse invariant defined by Plamenevskaya.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.