CAREER: COLORED LINK HOMOLOGIES AND THE GEOMETRIC TOPOLOGY OF SURFACES IN LOW-DIMENSIONS

Knots are one-dimensional substructures studied in interaction with the three-dimensional spaces that they live in, physically realized in coiled DNA and folded proteins in the micro, and in cosmic trajectories in the macro. The investigator will study a knot’s properties defined by its quantum invariants, constructed by summing over weighted “states” of the knot. The project aims to recover certain quantum invariants of a knot by assigning classical objects to each weight and state. This will, in a way, make precise the idea of quantum mechanical theories and classical theories predicting the same physical phenomena at large scale. In the long-term, the investigator will apply the results toward the goal of developing new theories governing the behavior of knots. For the broader impacts of the project, the investigator will recruit talented undergraduate students at Texas State University to work on summer research that directly contributes to the research program and train them to disseminate their work to the local community. The summer program will culminate in an annual quantum topology conference at Texas State University to support network development for early career mathematicians, that broadly includes graduate students and postdocs working in quantum topology in the United States. The project will proceed along three main lines of inquiry, where the first component investigates the simplification of a presentation of the Khovanov homology of torus links with the aim of giving a general decomposition of the colored Jones polynomial, the decategorification of colored Khovanov homology. With her collaborators the investigator will apply the formula for the decomposition to study the volume conjecture of a subfamily of highly-twisted alternating links. The second component will generalize Garoufalidis’ Jones slope conjecture to include surfaces in 4-dimensions and study the number-theoretical implications via Zeilberger’s machinery. The third direction of inquiry plans to recover and potentially construct new link and 3-manifold invariants from results of the first and second line of inquiries. 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.