We show that there exist infinitely many non-invertible, hyperbolic knots that admit toroidal Dehn surgery of hitting number four. The resulting toroidal manifold contains a unique incompressible torus meeting the core of the attached solid torus in four points, but no incompressible torus meeting it less than four points.
"Non-invertible knots having toroidal Dehn surgery of hitting number four." Hiroshima Math. J. 38 (3) 447 - 454, November 2008. https://doi.org/10.32917/hmj/1233152781