  We give a description of all (1,2)-knots in S^3 which admit a closed
meridionally incompressible surface of genus 2 in their complement. That is, we
give several constructions of (1,2)-knots having a meridionally incompressible
surface of genus 2, and then show that any such surface for a (1,2)-knot must
come from one of the constructions. As an application, we show explicit
examples of tunnel number one knots which are not (1,2)-knots.
