Explorations in Entanglement and Knotting in Low-Dimensional Topology

结和链接是三维环境中的闭合环，可能以有趣或复杂的配置包围。打结、连接和缠结在广泛的物理现象中发生。DNA在复制、重组和其他酶反应中可以连接和断开，或打结和解开。打结核酸的全局拓扑结构和酶的局部活性可以用来自结理论的对象和操作来建模。与此同时，纽结结构与三维和四维流形的关系在前沿几何学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。本项目从低维拓扑学的角度研究结、链接和缠结的理论和应用。该项目的潜在好处是我们对解结操作的理解的进步，揭示了链接和三流形的不变量之间的新关系，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更强大的数学框架。该奖项将通过支持结和链接不变量的在线数据库（KnotInfo）和弗吉尼亚联邦大学的系列讲座来提高数学素养并促进知识的广泛传播，该系列讲座在促进新兴研究主题的同时强调代表性不足的人和妇女的成就。该项目的中心焦点对象是结，链接和缠结的不变量。本研究使用Floer同调、Khovanov同调以及几何拓扑学中的技巧来探讨节点、链接与三流形之间的关系。第一个目标是解决结理论中关于交叉变化、缠结分解和解结的基本问题。新的解释Heegaard Floer和Khovanov理论不变量的缠结在浸没曲线的表面上的方法的一个重要组成部分。第二个目的是通过对Milnor不变量和Dehn手术的探索，研究链环不变量和三维流形之间的关系。第三个目标是揭示纽结理论与生物聚合物结构之间的新联系，并利用几何和低维拓扑学和图论的方法来探索Gordian型纽结图的广泛几何结构。该项目包括以空间θ曲线和核酸纠缠模型为中心的生物学动机问题。该项目为研究生和本科生提供了在理论和应用纽结理论方面做出贡献的途径。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。

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.