Mathematics

Dr. Abhijit Champanerkar
Department of Mathematics,College of Staten Island, CUNY

Abstract:  Knot theory has a fascinating history of using techniques  from diverse fields of mathematics. In this talk we will explore the
interactions between knot theory, graph theory and hyperbolic geometry. After giving some background I will focus on two natural
knot invariants, a geometric quantity called the volume density, and a diagrammatic quantity called the determinant density. We will
talk about recently discovered interesting relationships between the spectra of volume and determinant densities, and explore natural
questions and conjectures motivated by this study. The techniques used in answering some of these questions involve a blend of
geometry, topology, graph theory, dimer models, and the Mahler measure of two-variable polynomials. This is joint work with Ilya
Kofman and Jessica Purcell.