Lagrangians and Low-Dimensional Topology

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Topology is a mathematical field that seeks to understand the intrinsic properties of shape. This provides tools for many areas of science, including studying how enzymes knot up DNA or finding trends in large data sets. The aim of this project is to study the properties of knotted strings. One of the major goals is to understand mathematically how a knot changes if it is cut open and the ends are connected together differently, just as enzymes do to DNA. The prediction is that this always has a measurable change in the knottedness, and the PI seeks to verify this mathematically, enhancing understanding of the fundamental structure of knots. The PI's activities in this project, including mentorship and developing virtual mathematics communities, will create new educational opportunities for undergraduate and graduate students, with special focus on increasing diversity in and access to mathematics. The project will study key questions and structures in low-dimensional topology through a variety of invariants built from Lagrangian submanifolds. This includes using the description of knot Floer and Khovanov tangle invariants in terms of immersed Lagrangians in the four-punctured sphere to study the nugatory crossing conjecture, obtain lower bounds on the rank of knot homology theories in the presence of an essential Conway sphere, and prove detection results for the tangle invariants for certain classes of tangles. The PI will use the immersed Lagrangian invariants from bordered Heegaard Floer homology to give new bounds on the unknotting numbers of satellite knots. The PI will also use Lagrangians in the pillowcase arising from SU(2)-character varieties of knots towards the three-summands conjecture, which predicts that Dehn surgery on a knot in the three-sphere cannot consist of more than two prime summands. The PI will also mentor graduate students to develop new structural properties of symplectic instanton homology and to study the knot Floer homology of tunnel number one knots. In addition to organizing various seminars, the PI will assist in the development of the Floer homology open problem list and continue mentorship and other activities with the aim of promotion of underrepresented groups in mathematics.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.