Explorations in Entanglement and Knotting in Low-Dimensional Topology

低维拓扑中纠缠与打结的探索

• 批准号: 2204148
• 负责人: Allison Moore
• 金额: $ 29.46万
• 依托单位: Virginia Commonwealth University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204148&HistoricalAwards=false
• 关键词: Explorations; Entanglement; Knotting; Low; Dimensional

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.

结和链接在三维环境中是封闭环路，可能与有趣或复杂的配置交织在一起。打结，链接和纠缠发生在广泛的物理现象中。在复制，重组和其他酶促反应中，DNA可以链接，未连接，打结和无结。打结的核酸的全球拓扑结构和酶的局部活性可以用结理论对象和操作进行建模。同时，打结的结构与三维和四维流形的关系在领先的几何和拓扑中起着核心作用，最近的数学发展提供了新的工具，可以正式研究打结和链接。从低维拓扑的角度来看，该项目集中在结，链接和缠结的理论和应用上。该项目的潜在好处是我们对解开操作的理解，揭示了链接和三个manifolds之间不变的新关系，并为酶促活动和生物聚合物的塑性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节和链接不变性的在线数据库（Knotinfo）和弗吉尼亚州联邦大学的讲座系列来提高数学素养，并促进知识的广泛传播，并在促进新兴的研究主题，同时强调成就不足的人和女性的成就。在这个项目中，这个项目的重点不足。该研究使用几何拓扑中的浮动同源性，霍瓦诺夫同源性和技术来探索结，链接和三个manifolds之间的关系。第一个目的是在结理论中解决有关跨越变化，纠缠分解和脱节的基本问题。 Heegaard Floer和Khovanov理论不变的新解释是在表面上浸入曲线方面的主要解释。第二个目的是通过探索Milnor的不变式和Dehn手术来调查连接与三个manifolds之间的关系。第三个目的旨在发现结理论与生物聚合物结构之间的新联系，并探测Gordian型结图的广泛几何结构，并在几何学和低维拓扑和图理论中使用方法论。该项目包括以生物学动机的问题为中心，这些问题集中在空间theta曲线和核酸中的纠缠模型。该项目为研究生和本科生提供了为理论和应用结理论研究做出贡献的途径。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估审查标准，被认为值得通过评估。