LEAPS-MPS: Examination of knot invariants via computational methods

This project investigates knot theory; a field of mathematics concerned with understanding how loops (like knots in a string) can be embedded and manipulated in three-dimensional space. Knot theory is part of the broader area of topology, which studies the properties of shapes that remain unchanged when stretched or bent. While abstract in nature, knot theory plays a role in many scientific domains: molecular biology (where it helps explain DNA folding), physics (in the study of quantum entanglement), and computer science (in data analysis). The research will focus on mathematical objects called invariants: quantities or properties that remain unchanged when a knot is deformed in specific ways. These are crucial tools for distinguishing between different types of knots and understanding their structure. The project aims to uncover new patterns, propose novel tools for identification for certain families of knots, and push the boundaries of what is computationally accessible in topology. To advance this work, the project incorporates machine learning, a form of artificial intelligence where computers detect patterns in data, and predictive modeling, which uses data to make statistically informed guesses about unknown or complex systems. In addition, the PI will involve undergraduate students in active research and departmental programs, create computational packages to support further exploration, and engage with K–12 communities through outreach programs.

This work consists of several interrelated research topics, each addressing central questions in knot theory by bridging classical and modern Heegaard Floer-theoretic perspectives with use of computational methods. With the help of machine learning, the PI will uncover new relationships between the Alexander polynomial and various Floer-theoretic invariants, examining how L-space knots behave in the knot concordance group. The PI also aims to develop a new invariant which is expected to generalize, in some sense, both the Upsilon function and the Upsilon torsion function. Using related techniques combined with classical study of covers of knots, the PI and her collaborator will develop prime knot detection results. In work with her collaborators, the exploration of null-homologous twisting will produce generalizations of known results for the unknotting number.

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.

Award Number: 2532488
Principal Investigator: Samantha Allen
Funds Obligated: $249,941
State: PA