Higher-order Phenomena in Knot Theory

结理论中的高阶现象

• 批准号: 1205922
• 负责人: Peter Horn
• 金额: $ 13.2万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-15 至 2012-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205922&HistoricalAwards=false
• 关键词: Higher; order; Phenomena; Knot; Theory

The principal investigator will study knots up to isotopy and up to concordance using a combination of techniques from the categories of topological and smooth manifolds, including von Neumann rho-invariants and Heegaard Floer homology. The primary goal of this project is to determine structure in the group of smooth concordance classes of topologically slice knots. Using grid diagrams, the principal investigator will work to find a refinement of knot Floer homology which hopefully reflects the non-commutative structure of the fundamental group of the knot exterior. From this refinement, the principal investigator hopes to garner further concordance invariants and apply them to study topologically slice knots. This refinement of knot Floer homology is expected to capture some of the information in the higher-order Alexander modules. The final goal of the project is to develop computational techniques for these modules.The principal investigator hopes to shed light on the structure of knotted curves in space. Knots appear many places in nature. For example, the strands of DNA in cells are knotted, and these knots can affect the replication process. Many of the existing algebraic tools used to study knots are commutative. The principal investigator will develop new non-commutative algebraic tools to study knots and will develop techniques for making computations. By using more general, non-commutative algebraic techniques, the principal investigator hopes to reveal more structure in the set of knots in space.

主要研究员将研究结的合痕和一致性，使用拓扑和光滑流形，包括冯诺依曼rho不变量和Heegaard Floer同源性的技术组合。 这个项目的主要目标是确定拓扑切片结的光滑协调类组中的结构。 使用网格图，主要研究人员将努力找到一个完善的结弗洛尔同源，希望反映了非交换结构的基本组的结外部。 通过这种改进，首席研究员希望获得进一步的一致性不变量，并将其应用于拓扑切片结的研究。 这种改进的结弗洛尔同源性，预计捕捉一些信息，在高阶亚历山大模块。 该项目的最终目标是为这些模块开发计算技术。首席研究员希望阐明空间中打结曲线的结构。 结在自然界中出现在许多地方。 例如，细胞中的DNA链是打结的，这些结会影响复制过程。 许多用于研究结的现有代数工具是可交换的。 首席研究员将开发新的非交换代数工具来研究结，并将开发计算技术。 通过使用更一般的，非交换代数技术，首席研究员希望揭示更多的结构在空间中的节点集。