RUI: Topological Characterizations of Generalized Alternating Knots

Vassar College, Poughkeepsie NY

Investigators

Abstract

The main topic of this National Science Foundation funded research project lies in low-dimensional topology and knot theory. Topology is the study of mathematical spaces up to elastic stretching, and knot theory is the study of how loops can be embedded into topological spaces. Knot theory and low-dimensional topology address important problems in physics, chemistry, and biology. In addition to its research goals, this project has an educational focus. Several of the proposed research projects are intended for collaboration with undergraduate students. This project also aims to make knot theory accessible to a broad audience via mentoring and outreach programs involving audiences that are not typically exposed to research mathematics. Connecting combinatorial information describing a topological object with the underlying topology or geometry is a common theme throughout topology. Despite being defined in a combinatorial manner, alternating knots and links are now known to have topological descriptions. These topological descriptions led to a recognition algorithm for alternating knots as well as a new proof of a fundamental theorem concerning alternating links. In this project, the PI plans to expand these techniques to several classes of links that generalize alternating links, including quasi-alternating links, almost alternating links, links of Turaev genus one, and links with simple alternating tangle decompositions. The Jones polynomial and Khovanov homology of links in these classes will also be explored. 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.