Extending fibrations of knot complements to ribbon disk complements

Abstract

We show that if K is a fibered ribbon knot in S^3=(boundary B^4) bounding ribbon disk D, then with a transversality condition the fibration on the complement of K in S^3 extends to a fibration of the complement of D in B^4. This partially answers a question of Casson and Gordon (``If D is a homotopy-ribbon disk bounded by a fibered knot, is D fibered?"). In particular, we show the fibration always extends when D has exactly two local minima, extending a result of Scharlemann for the unknot. More generally, we construct movies of singular fibrations on 4-manifolds and describe a sufficient property of a movie to imply the underlying 4-manifold is fibered over S^1.