  We consider irreducible 3-manifolds M that arise as knot complements in
closed 3-manifolds and that contain at most two connected strict essential
surfaces. The results in the paper relate the boundary slopes of the two
surfaces to their genera and numbers of boundary components. Explicit
quantitative relationships, with interesting asymptotic properties, are
obtained in the case that M is a knot complement in a closed manifold with
cyclic fundamental group.
