RUI:Hyperbolic 3-Manifolds and Knots

Williams College, Williamstown MA

Investigators

Abstract

In 1978, William Thurston revolutionized low-dimensional topology by demonstrating that many 3-manifolds have hyperbolic metrics. The Mostow Rigidity Theorem then implies that these metrics can be used to generate interesting invariants to distinguish hyperbolic 3-manifolds. A tremendous number of results have come out of these ideas. The goal of this proposal is to further understand hyperbolic 3-manifolds. In particular, there are a set of invariants that come out of the cusps of hyperbolic 3-manifolds, and hyperbolic knots, including the cusp volume, the meridian length and the longitude length. The plan is to continue to work on these invariants, delineating better their possible range of values and determining their implications for the rest of the hyperbolic structure. Moreover, recent work has demonstrated the existence of totally geodesic surfaces in hyperbolic knot complements. Their existence is suprising and exciting. It is hoped that a determination can be made as to which knots possess such surfaces. Understanding hyperbolic 3-manifolds has implications for determining the shape of the spatial universe. These invariants will be the kind that cosmologists may use to empirically identify the universe. Although recent measurements of the cosmic background radiation point to a curvature very close to flat, this could mean that the universe is hyperbolic, but very large. Then more subtle hyperbolic invariants will be necessary to determine its topological shape.