## 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.