Smallish knots and triangulations
by
Marc Culler
University of Illinois at Chicago
Coauthors: Nathan Dunfield, William Jaco, Peter B. Shalen

A knot in a closed orientable non-Haken 3-manifold is smallish if the complement is irreducible and contains no essential surface with meridian boundary curves. It would follow from a difficult conjecture due to Lopez that non-Haken manifolds contain smallish knots. There are interesting connections between this weak version of the Lopez conjecture and the Poincaré conjecture. In many cases smallish knots arise as edges in 1-vertex triangulations. We will discuss a result which shows that certain simplification operations are possible when no edge of a 1-vertex triangulation of a non-Haken manifold is a smallish knot.

Date received: February 9, 2001