Efficient triangulations and boundary slopes

Abstract

For a compact, orientable, irreducible, ∂-irreducible, and an-annular 3-manifold, it is shown there are only finitely many boundary slopes for incompressible and ∂-incompressible surfaces of a bounded Euler characteristic. We use normal surface theory and the inverse relationship of crushing a triangulation along a normal surface [8] and that of inflating an ideal triangulation [12] to introduce and study boundary-efficient triangulations and end-efficient ideal triangulations. It is shown for a compact 3-manifold with boundary, satisfying these topological conditions, any triangulation can be modified to a boundary-efficient triangulation; furthermore, it can be decided if a triangulation of such a manifold is boundary-efficient.