Let K be a knot in S3, and
consider incompressible (in the stronger sense of π1-injective), ∂-incompressible
surfaces S in the exterior of K. A question which has been around for some time is
whether the boundary-slope function S↦mS∕lS, where mS and lS are the numbers of
times each circle of ∂S wraps around K meridionally and longitudinally, takes on
only finitely many values (for fixed K). This is known to be true for certain knots:
torus knots, the figure-eight knot [4], 2-bridge knots [2], and alternating knots
[3]. In this paper an affirmative answer is given not just for knot exteriors,
but for all compact orientable irreducible 3-manifolds M with ∂M a torus.
Further, we give a natural generalization to the case when ∂M is a union of
tori.
