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? 



Looking for examples of large hyperbolic two-generator knots or 3-manifolds
Mario Eudave-Muñoz constructed examples of tunnel number one hyperbolic knots containing closed incompressible surfaces. See Theorem 8.1: 


*Mario Eudave-Muñoz*, MR 1719999 [**Incompressible surfaces in tunnel number one knot complements**](http://dx.doi.org/10.1016/S0166-8641(99)00040-1), *Topology Appl.* **98** (1999), no. 1-3, 167--189.


