Knot and Link Concordance

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Topology is the study of the shape of spaces, called manifolds, that locally look like Euclidian space but globally can be quite complicated or curved. For example, the surface of the earth or the surface of a donut are examples of manifolds. In this case, on the surface there are two directions to move in and hence they are called 2-dimensional manifolds. This award will be used to study the topology of manifolds in dimensions 3 and 4 as well as how the natural submanifolds in both dimensions interact. The study of manifolds in dimensions 3 and 4 are of utmost important in science as it uncovers information about the world we live in (3-dimensions) as well as 4-dimensional objects which can be seen as space-time. The main goal of this project is to better understand which knots and links can be the slice of sphere in 3-dimensional space as it moves through space-time, called slice knots and links. This is especially important in studying the topology of 4-dimensional manifolds as it relates to one of the most important questions in topology: the Smooth 4-dimensional Poincare Conjecture. This conjecture says that if one can compute certain information about a 4-dimensional manifold and one gets the same answer as for the standard 4-dimensional sphere, then it is actually a sphere. This is the only dimension where it is still unknown. In addition, the PI aims to make progress to the famous Slice-ribbon Conjecture, which says that if a knot is slice then we can deform the 4-sphere in space-time so that is it "nice". Understanding topology and methods for homological algebra is especially important at this time as it is crucial to the study of topological data analysis, a leading area of data science. The PI will present the work at conferences, as well as publish and put the article on the public repository, arxiv.org so that the results are available to all. The PI will run conferences and special sessions where the majority of speakers are women or other underrepresented areas of mathematics. The award provides funds to train graduate students in research.Knots and links are of fundamental importance in the study of 3- and 4-dimensional manifolds. For example, we can present any 4-manifold as a Kirby diagram of weighted links. Furthermore, topological knot and link concordance has a strong relationship to Freedman's surgery-theoretic scheme for classifying 4-dimensional manifolds. While much is understood about 3-manifolds, because of the geometrization conjecture and work of Agol, the study of smooth 4-dimensional manifolds is still poorly understood. This project will further our knowledge of knot and links concordance, as well as general 3- and 4-dimensional topology, subjects of vital importance in mathematics. The goal of the project is to have a better understanding of knot and link concordance as well as questions about 3-manifold groups. To do this, the PI will study the algebraic structure of the knot and links concordance groups along with their filtrations like the n-solvable and bipolar filtrations. Related to this, the PI will study their various metric spaces to show that they have the structure of a fractal space. In addition, the PI will study the behavior of knot and string link satellite operators on the space.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.