Surfaces and Geometry and Topology of Quantum Link Invariants

结理论研究三维空间中被认为是连续变换的打结的闭合环。在数学中，如果不切断和调整形成环的弦就不能转换成另一个结，就被认为是彼此不同的，或者换句话说，是不相等的。区分不等价结点的关键工具叫做结点不变量这些本质上是由空间的连续变换保留的结点的性质。在这个项目中，PI将研究量子结不变量之间的关系，量子结不变量是一个相对较少被理解的家族，由量子物理和数学的思想构建而成。量子拓扑学的核心问题与三维空间的几何拓扑学、由量子不变量赋值的多项式的数论性质以及定义量子拓扑学理论的量子群的代数结构有着广泛的联系。本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三流形拓扑和法向曲面理论的技术，扩展定义彩色琼斯多项式的状态和与链路补中适当嵌入曲面之间的对应关系。下一部分是证明彩色Khovanov同调的强斜率猜想的一个分类版本，它是彩色Jones多项式的一种分类。PI将评估该框架的潜力，以提供关于Khovanov同调与knot flower同调关系的新视角，这是另一个已被广泛研究并与其他领域有着深刻联系的链接不变量。本项目将进一步研究Khovanov同调的稳定性。这些结果将用于探索由Plamenevskaya定义的横向不变量的接触几何性质。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.