Georgia Institute of Technology College of Sciences

Lickorish observed a simple way to make two knots in S^3 that produced the same manifold by the same surgery. Many have extended this result with the most dramatic being Osoinach's method (and Teragaito's adaptation) of creating infinitely many distinct knots in S^3 with the same surgery yielding the same manifold. We will turn this line of inquiry around and examine relationships within such families of corresponding knots in the resulting surgered manifold.

We prove that every metric of constant curvature on a compact 2-manifold M with boundary bdM induces (at least) four vertices, i.e., local extrema of geodesic curvature, on bdM, if, and only if, M is simply connected. Indeed, when M is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant curvature on M which induce only two vertices on bdM. Furthermore, we characterize the sphere as the only closed orientable Riemannian 2-manifold M which has the four-vertex-property, i.e., the boundary of every compact surface immersed in M has 4 vertices.

In this talk I will describe some relations between embedded graphs, their polynomials and the Jones polynomial of an associated link. I will explain how relations between graphs, links and their polynomials leads to the definition of the partial dual of a ribbon graph. I will then go on to show that the realizations of the Jones polynomial as the Tutte polynomial of a graph, and as the topological Tutte polynomial of a ribbon graph are related, surprisingly, by the homfly polynomial.

W. Goldman proved that the SL(2)-character variety X(F) of a closed surface F is a holonomic symplectic manifold. He also showed that the Sl(2)-characters of every 3-manifold with boundary F form an isotropic subspace of X(F). In fact, for all 3-manifolds whose SL(2)-representations are easy to analyze, these representations form a Lagrangian space. In this talk, we are going to construct explicit examples of 3-manifolds M bounding surfaces of arbitrary genus, whose spaces of SL(2)-characters have dimension as small as possible. We discuss relevance of this problem to quantum and classical low-dimensional topology.