RUI:Hyperbolic 3-Manifolds and Knots

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In 1978, William Thurston revolutionized low-dimensional topology bydemonstrating that many 3-manifolds have hyperbolic metrics. The MostowRigidity Theorem then implies that these metrics can be used to generateinteresting invariants to distinguish hyperbolic 3-manifolds. A tremendousnumber of results have come out of these ideas. The goal of this proposalis to further understand hyperbolic 3-manifolds. In particular, there are aset of invariants that come out of the cusps of hyperbolic 3-manifolds, andhyperbolic knots, including the cusp volume, the meridian length and thelongitude length. The plan is to continue to work on these invariants,delineating better their possible range of values and determining theirimplications for the rest of the hyperbolic structure. Moreover, recentwork has demonstrated the existence of totally geodesic surfaces inhyperbolic knot complements. Their existence is suprising and exciting. Itis hoped that a determination can be made as to which knots possess suchsurfaces.Understanding hyperbolic 3-manifolds has implications for determining theshape of the spatial universe. These invariants will be the kind thatcosmologists may use to empirically identify the universe. Although recentmeasurements of the cosmic background radiation point to a curvature veryclose to flat, this could mean that the universe is hyperbolic, but verylarge. Then more subtle hyperbolic invariants will be necessary todetermine its topological shape.