Topology and Geometry of the Jones Polynomial via Skein Algebras

This project investigates important problems in quantum topology and its deep connections with classical topology and hyperbolic geometry. Quantum topology focuses on the study and classification of invariants of three- and four-dimensional spaces and the knotted circles they contain. These structures naturally arise in various scientific contexts, including DNA modeling and theoretical physics, and have numerous applications. Techniques and methods from quantum topology may also contribute to the development of both theoretical and practical models for quantum computation. The project is inherently interdisciplinary, drawing on ideas and methods from topology, geometry, algebra, number theory, analysis, quantum field theory, and combinatorics. It also emphasizes the mentoring and training of students and postdoctoral researchers. The Principal Investigator (PI) will concentrate on three closely related research directions. The first is the AJ conjecture, which relates the colored Jones and HOMFLYPT polynomials to the fundamental group of knots. The second involves the development and exploration of hyperbolic topological quantum field theory, with the goal of advancing the volume conjecture and resolving the AJ conjecture. The third focuses on stated skein algebras of surfaces, which have wide-ranging applications, including a potential partial proof of the duality conjecture in higher Teichmüller theory. 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.