Higher-order Phenomena in Knot Theory

首席研究员将使用拓扑流形和光滑流形的组合技术，包括冯·诺伊曼不变量和Heegaard Floer同调，研究到同位素和一致性的结。本项目的主要目标是确定拓扑切片结的光滑一致性类组中的结构。利用网格图，首席研究员将努力找到一个改进的结花同源性，希望反映结外部基本群的非交换结构。通过这种改进，首席研究员希望获得更多的一致性不变量，并将其应用于拓扑切片结的研究。这种对结花同源性的改进有望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。结在自然界中出现的地方很多。例如，细胞中的DNA链是打结的，这些打结会影响复制过程。许多现有的用于研究结点的代数工具都是可交换的。首席研究员将开发新的非交换代数工具来研究结，并将开发进行计算的技术。通过使用更一般的、非交换的代数技术，首席研究员希望揭示空间中结的更多结构。

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.