  We consider closed acylindrical surfaces in 3-manifolds and in knot and link
complements, and show that the genus of these surfaces is bounded linearly by
the number of tetrahedra in the triangulation of the manifold and by the number
of rational (or alternating) tangles in a projection of a link (or knot). For
each g we find knots with tunnel number 2 and manifolds of Heegaard genus 3
containing acylindrical surfaces of genus g. Finally, we construct 3-bridge
knots containing quasi-Fuchsinan surfaces of unbounded genus, and use them to
find manifolds of Heegaard genus 2 and homology spheres of Heegaard genus 3
containing infinitely many incompressible surfaces.
