Explorations in Entanglement and Knotting in Low-Dimensional Topology

结和链接是三维环境中的闭合环，可能缠绕在有趣或复杂的配置中。打结、连接和纠缠发生在广泛的物理现象中。在复制、重组和其他酶反应中，DNA可以连接和非连接，或打结和解结。打结核酸的全局拓扑结构和酶的局部活性可以用纽结理论中的对象和操作来模拟。同时，纽结结构与三维和四维流形的关系在前沿几何和拓扑学中起着核心作用，最近的数学发展为正式研究纽结和链接提供了新的工具。本课题主要从低维拓扑学的角度研究纽结、链环和缠结的理论和应用。这个项目的潜在好处是在我们对解结操作的理解方面取得了进展，揭示了链接不变量和三维流形之间的新关系，并为生物聚合物的酶活性和拓扑结构的建模和分析提供了一个更强大的数学框架。该奖项将通过支持纽结和链接不变量的在线数据库(KnotInfo)和弗吉尼亚联邦大学的一系列讲座来提高数学素养和促进知识的广泛传播，该系列讲座在强调未被充分代表的人和妇女的成就的同时促进新兴研究主题。这个项目的中心焦点是纽结、链接和缠结的不变量。该研究利用Floer同调、Khovanov同调和几何拓扑学中的技巧来探索纽结、链环和三种流形之间的关系。第一个目标是解决纽结理论中关于交叉变换、缠结分解和解结的基本问题。对Heegaard Floer和Khovanov理论不变量的新解释是该方法的一个主要组成部分。第二个目的是通过对Milnor不变量和Dehn手术的探讨，研究环的不变量与三维流形之间的关系。第三个目的旨在揭示纽结理论与生物聚合物结构之间的新联系，并用几何和低维拓扑学和图论的方法论来探索Gordian类型纽结图的广泛几何结构。该项目包括生物动机的问题，这些问题集中在空间θ曲线和核酸纠缠模型上。该项目为研究生和本科生提供了为理论和应用纽结理论研究做出贡献的途径。该奖项反映了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。

Knots and links are closed loops in a 3-dimensional environment, possibly entwined in an interesting or complicated configuration. Knotting, linking, and entanglement occur in a broad range of physical phenomena. DNA can become linked and unlinked, or knotted and unknotted, during replication, recombination, and in other enzymatic reactions. The global topology of knotted nucleic acids and the local activity of enzymes can be modeled with objects and operations from knot theory. At the same time, the relationship of knotted structures with three and four-dimensional manifolds plays a central role in leading-edge geometry and topology, where recent mathematical developments have provided new tools with which to formally investigate knotting and linking. This project is centered on the theory and applications of knots, links, and tangles from the perspective of low-dimensional topology. Potential benefits of this project are advances in our understanding of unknotting operations, uncovering new relationships between invariants of links and three-manifolds, and providing a more robust mathematical framework for the modeling and analysis of enzymatic activities and topological structures of biopolymers. This award will increase mathematical literacy and promote broad dissemination of knowledge by supporting an online database of knot and link invariants (KnotInfo) and a lecture series at Virginia Commonwealth University that promotes emerging research topics while emphasizing achievements of underrepresented people and women.The central objects of focus in this project are invariants of knots, links, and tangles. The research uses Floer homology, Khovanov homology, and techniques in geometric topology to explore the relationships between knots, links, and three-manifolds. The first aim is to resolve fundamental questions in knot theory on crossing changes, tangle decompositions, and unknotting. New interpretations of Heegaard Floer and Khovanov-theoretic invariants of tangles in terms of immersed curves on surfaces are a major component of the methodology. The second aim is to investigate the relationship between invariants of links and three-manifolds through an exploration of Milnor's invariants and Dehn surgery. The third aim seeks to uncover new connections between knot theory and the structure of biopolymers, and to probe the broad geometric structure of Gordian-type knot graphs with methodology in geometric and low-dimensional topology and graph theory. The project includes biologically motivated questions that center on spatial theta-curves and models of entanglement in nucleic acids. The project provides avenues for graduate and undergraduate students to contribute to research in theoretical and applied knot theory.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.