We say a knot $K$ in $S^3$ is small if its exterior contains no closed properly embedded incompressible surfaces and we say $K$ is large otherwise.

Does anyone know of an example of a large hyperbolic knot $K$ such that $\pi_1( S^3 - K)$ has rank 2? All of the examples of large knots I can think of (4-strand pretzel knots and some knots whose complements contain embedded quasi-Fuchsian surfaces for instance) seem to have fundamental groups with rank 3 or larger.

How about, more generally, examples of large hyperbolic 3-manifolds with rank two fundamental group?