CDS&E: RUI: Collaborative Research: Data-driven methods in classical knot theory

California State University-Long Beach Foundation, Long Beach CA

Investigators

Abstract

The philosophy driving this mathematics research project is to approach the study of knots from a data-driven perspective. The project aims to employ powerful computational techniques to calculate invariants on a large scale, make these algorithms and their output available to the community, and use computational resources to empirically arrive at and test conjectures. The investigators plan to develop new algorithms that will expand existing databases of knot invariants and may lead to resolving old open problems, thus also broadening the potential for applications throughout the sciences, from molecular biology to quantum physics. Mathematical tools for studying knots generally fall into two categories: geometric and algebraic. The project will rely on large-scale computations to analyze connections between these two points of view, harnessing the data collected to attack questions that have so far remained intractable. In addition, the project intends to make deep questions in the field accessible to students by developing new combinatorial and exploratory techniques. The investigators will actively engage students in impactful research, directly addressing known pipeline gaps in the field of mathematics. The research goals of this project include: (1) approaching the meridional rank conjecture from a computational perspective, with the aim of verifying the conjecture for all tabulated knots and extracting theoretical results from these empirical findings; (2) designing an efficient algorithm to compute bridge numbers for large classes of knots by combining techniques for finding lower and upper bounds for these numbers from knot diagrams; (3) computing homotopy ribbon obstructions for slice knots using Kjuchukova's invariants and developing algorithms to test potential counterexamples to the slice-ribbon conjecture; (4) constructing four-manifolds as branched covers of the sphere with knots as singularities on the branching sets; classifying these branched covers and studying, with the help of trisections, the smooth structures on the four-manifolds constructed. 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.