Surfaces and Geometry and Topology of Quantum Link Invariants

纽结理论研究三维空间中的打结、闭合环，被认为是空间的连续变换。在数学中，不切断和调整形成环的弦就不能相互转换的结被认为是彼此不同的，换句话说，是不等价的。区分不等价纽结的关键工具称为纽结不变量，这些本质上是通过空间的连续变换保持的纽结的性质。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。量子拓扑学中的核心问题广泛地与三维空间的几何拓扑、由量子不变量赋值的多项式的数论性质以及定义该理论的量子群的代数结构有关。该项目包括本科生和硕士研究生的研究计划。国际扶轮还将参与“编程女孩”俱乐部会议，为当地社区服务。有色Jones多项式是一种重要的纽结不变量，它源于量子群的表示理论，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维流形拓扑和法线曲面理论来扩展定义有色Jones多项式的状态和和适当嵌入在链补中的曲面之间的对应关系。下一部分证明了有色Khovanov同调的强Slope猜想的范畴化形式，它是有色Jones多项式的范畴。PI将评估这一框架的潜力，以提供关于Khovanov同调与纽结Floer同调的关系的新视角，纽结Floer同调是另一个已被广泛研究的链接不变量，与其他领域有着深刻的联系。该项目将进一步研究Khovanov同调的稳定性性质。这些结果将被用来探索由Plamenevskaye定义的横向不变量的接触几何性质。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.