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A knot k in a closed orientable 3-manifold is called nonsimple if theexterior of k possesses a properly embedded essential surface of nonnegativeEuler characteristic. We show that if k is a nonsimple prime tunnel number oneknot in a lens space M (where M does not contain any embedded Klein bottles),then k is a (1,1) knot. Elements of the proof include handle addition and Dehnfilling results/techniques of Jaco, Eudave-Munoz and Gordon as well asstructure results of Schultens on the Heegaard splittings of graph manifolds.

On nonsimple knots in lens spaces with tunnel number one