The Lopez Conjecture

This project brings together M. Culler, N. Dunfield, W. Jaco, D. Letscher, H. Rubinstein, and P. Shalen to study essential surfaces in knot exteriors. This project will involve such a study for knots in a closed, orientable 3-manifold where we are seeking to understand the topology of the manifold. More specifically, we will take up the program of Culler and Shalen which attempts to identify properties of a knot in a 3-manifold, in terms of essential surfaces in the knot exterior, which lead to strong conclusions about the topology of the 3-manifold. We say such properties detect a knot type.

Through their study of the PSL(2,C) character variety of the fundamental group of a one cusp hyperbolic 3-manifold, Culler and Shalen obtain deep results leading to properties which detect the trivial knot in a closed, orientable 3-manifold with cyclic fundamental group (a knot is trivial if it has a nonseparating disk properly embedded in its exterior). On the other hand, having identified properties which detect certain knot types, one must have, under some reasonable restrictions on the 3-manifold, that there exists a knot in the manifold having the particular properties. If this can be done, we say the property is realizable in 3-manifolds satisfying the restrictions. For example, if the property that detects the trivial knot in closed, orientable, irreducible 3-manifolds with cyclic fundamental group is also realizable in such manifolds, then these manifolds are lens spaces; in particular, a simply connected one is the 3-sphere (The Poincare Conjecture).

While the methods of Culler and Shalen have been very successful at identifying properties which detect knot types, these methods are not designed to show that a property is realizable. Recent work of W. Jaco and H. Rubinstein on efficient triangulations provides new, hopeful techniques for the realizability side of this program. Jaco and Rubinstein have shown that under reasonable restrictions a 3-manifold admits a triangulation in which each edge is a knot (one vertex triangulation) and particular edges, ``thick edges," are candidates for realizing knots with the desired properties, depending on the initial restrictions on the 3-manifold. A major step in this project, and a central test of the Jaco and Rubinstein techniques, is to show that for a closed, orientable, irreducible, nonHaken 3-manifold there is a triangulation where each edge is a knot and for some edge there is no closed, essential surface in the knot exterior. We say such a knot is a small knot. Therefore, we wish to show that a small knot can be realized in such 3-manifolds. This is the Lopez Conjecture.