Legendrian graphs, invariants and DNA topology

传奇图、不变量和 DNA 拓扑

• 批准号: 1406481
• 负责人: Danielle O'Donnol
• 金额: $ 14.46万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2015-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406481&HistoricalAwards=false
• 关键词: Legendrian; graphs; invariants; DNA; topology

The goal of this project is to broaden our understanding of many topological aspects of spatial graphs, and make it possible to answer larger questions about its external three-dimensional space (or three-space) and biological mechanisms. A graph is a set of points called vertices and curves between the points called edges. A spatial graph is a graph that is sitting in three-space. A spatial graph can be very complicated and knotted. Spatial graph theory is a recent field which grew out of knot theory, the study of knotted circles in three-space. In addition to contributing to our understanding of connections between spatial graphs and their surrounding three-space, this project has direct applications to biological questions. Knot theory has become important within the life sciences, in the field of DNA Topology. Most of the DNA that is worked with in laboratories is in small circular molecules that are called plasmids. Circular DNA also appears in nature. Plasmids can naturally be modeled with knots. With every cell division the DNA is copied in a process called replication. This is a complicated process that involves a number of enzymes. After replication the two new plasmids are linked (or catenated) and sometimes knotted. The reason for knotting is not completely understood. In the middle of replication DNA forms the more complicated structures, that of a knotted graph, this is an area of DNA topology that has yet to be investigated by mathematicians. This project will have immediate contributions to our biological understanding of DNA replication. In particular, the PI will focus on three main projects centering around spatial graphs: First, the PI will study a number of aspects of Legendrian graphs. Central results in contact geometry use Legendrian graphs in their proofs. Thus this opens an exciting new area with great potential. These projects will increase the overall understanding of Legendrian graphs, further increasing their potential to use Legendrian graphs as a tool to understand 3-manifolds, and potentially distinguish contact structures. Second, the PI will develop new invariants and better understand existing invariants of spatial graphs and knots, including graph Floer homology. Third, the PI will begin classification of unknotting number one theta-graphs to understand unknotting in DNA replication intermediates.

该项目的目标是扩大我们对空间图的许多拓扑方面的理解，并使其有可能回答有关其外部三维空间（或三维空间）和生物机制的更大问题。 图是一组称为顶点的点和称为边的点之间的曲线。 空间图是一个位于三维空间中的图。 空间图可以非常复杂和打结。 空间图论是从纽结理论发展而来的一个新领域，纽结理论研究三维空间中的纽结圈。 除了有助于我们理解空间图形及其周围三维空间之间的联系外，该项目还直接应用于生物学问题。 纽结理论在生命科学中，在DNA拓扑学领域中变得重要。 在实验室中，大多数DNA都是小的环状分子，称为质粒。 自然界中也存在环状DNA。 自然地，塑性体可以用结来建模。 每一次细胞分裂，DNA都在一个被称为复制的过程中被复制。 这是一个复杂的过程，涉及许多酶。 复制后，两个新质粒连接（或连锁），有时打结。 打结的原因还不完全清楚。 在复制过程中，DNA形成了更复杂的结构，即打结图，这是DNA拓扑学的一个领域，尚未被数学家研究。 该项目将对我们对DNA复制的生物学理解做出直接贡献。 特别是，PI将专注于围绕空间图的三个主要项目：首先，PI将研究Legendrian图的许多方面。 接触几何中的中心结果在它们的证明中使用勒让德图。因此，这开辟了一个具有巨大潜力的令人兴奋的新领域。这些项目将增加对Legendrian图的整体理解，进一步增加其使用Legendrian图作为理解3-流形的工具的潜力，并可能区分接触结构。 其次，PI将开发新的不变量，并更好地理解空间图和节点的现有不变量，包括图Floer同源性。 第三，PI将开始对解结一号θ图进行分类，以了解DNA复制中间体中的解结。