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.

首席研究者将使用来自拓扑和光滑歧管类别的技术组合（包括von Neumann Rho-Invariants和Heegaard Floer同源性）来研究与同位素的结合和一致性。 该项目的主要目标是确定拓扑切片结的平滑一致性类别的结构。 使用网格图，主要研究者将努力找到结式浮点同源性的改进，这希望反映了结外部基本组的非交通性结构。 通过这种改进，主要研究人员希望能够获得进一步的一致性不变性，并将其应用于拓扑结。 预计这种结浮子同源性的这种完善将捕获高阶亚历山大模块中的一些信息。 该项目的最终目标是开发这些模块的计算技术。主要研究人员希望阐明太空中打结曲线的结构。 结出大自然的许多地方。 例如，打结了细胞中DNA的链，这些结会影响复制过程。 用于研究结的许多现有代数工具都是可交换的。 首席研究者将开发新的非交通性代数工具来研究结，并将开发用于计算的技术。 通过使用更一般的非交互代数技术，主要研究人员希望在太空中的一组结中揭示更多的结构。