This is an expository talk. It is well-known that any hyperbolic
knot of genus one does not admit 2-string essential tangledecompositions.
In this talk we focus on essential tangle decompositions of knots with
crosscap number two or three. As anapplication we give a short proof that
the ribbon number of the Kinoshita-Terasaka knot is three. How about Conway's
knots?