NSF Award
2204148
Knots and links are closed loops in a 3-dimensional environment, possibly entwined in an interesting or complicated configuration. Knotting, linking, and entanglement occur in a broad range of physical phenomena. DNA can become linked and unlinked, or knotted and unknotted, during replication, recombination, and in other enzymatic reactions. The global topology of knotted nucleic acids and the local activity of enzymes can be modeled with objects and operations from knot theory. At the same time, the relationship of knotted structures with three and four-dimensional manifolds plays a central role in leading-edge geometry and topology, where recent mathematical developments have provided new tools with which to formally investigate knotting and linking. This project is centered on the theory and applications of knots, links, and tangles from the perspective of low-dimensional topology. Potential benefits of this project are advances in our understanding of unknotting operations, uncovering new relationships between invariants of links and three-manifolds, and providing a more robust mathematical framework for the modeling and analysis of enzymatic activities and topological structures of biopolymers. This award will increase mathematical literacy and promote broad dissemination of knowledge by supporting an online database of knot and link invariants (KnotInfo) and a lecture series at Virginia Commonwealth University that promotes emerging research topics while emphasizing achievements of underrepresented people and women.<br/><br/>The central objects of focus in this project are invariants of knots, links, and tangles. The research uses Floer homology, Khovanov homology, and techniques in geometric topology to explore the relationships between knots, links, and three-manifolds. The first aim is to resolve fundamental questions in knot theory on crossing changes, tangle decompositions, and unknotting. New interpretations of Heegaard Floer and Khovanov-theoretic invariants of tangles in terms of immersed curves on surfaces are a major component of the methodology. The second aim is to investigate the relationship between invariants of links and three-manifolds through an exploration of Milnor's invariants and Dehn surgery. The third aim seeks to uncover new connections between knot theory and the structure of biopolymers, and to probe the broad geometric structure of Gordian-type knot graphs with methodology in geometric and low-dimensional topology and graph theory. The project includes biologically motivated questions that center on spatial theta-curves and models of entanglement in nucleic acids. The project provides avenues for graduate and undergraduate students to contribute to research in theoretical and applied knot theory.<br/><br/>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.
