Efficient triangulations and boundary slopes

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