Surfaces and Geometry and Topology of Quantum Link Invariants

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Knot theory studies knotted, closed loops in a three-dimensional space considered up to continuous transformations of the space. In mathematics, knots that cannot be transformed into one another without cutting and regluing the strings that form the loops are considered to be different from each other, or in other words, inequivalent. Key tools for distinguishing inequivalent knots are called knot invariants and these are essentially the properties of a knot that are preserved by continuous transformations of the space. In this project the PI will study the relationship between quantum knot invariants, a relatively less understood family, constructed from the ideas of quantum physics and mathematics. Central questions in quantum topology are broadly connected to the geometric topology of three-dimensional spaces, number-theoretic properties of polynomials assigned by quantum invariants, and the algebraic structures of quantum groups defining the theory. The project includes research plans for undergraduate and Master's students. The PI will also engage with "Girls Who Code" club meetings in service to the local community. The colored Jones polynomial is an important knot invariant that comes from the representation theory of quantum groups and lies at the heart of quantum topology, low-dimensional topology, and hyperbolic geometry. In this project the PI and her collaborators will expand the correspondence between a state sum defining the colored Jones polynomial and properly embedded surfaces in a link complement, by applying the techniques of classical three-manifold topology and normal surface theory. The next part is to prove a categorified version of the Strong Slope Conjecture for colored Khovanov homology, which is a categorification of the colored Jones polynomial. The PI will evaluate the potential for this framework to provide new perspectives on the relationship of Khovanov homology to knot Floer homology, another link invariant that has been extensively studied with deep connections to other fields. The project will further study the stability properties of Khovanov homology. These results will be used to explore the contact-geometric properties of the transverse invariant defined by Plamenevskaya.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.