Extending fibrations on knot complements to ribbon disk complements

Abstract

We show that if $K$ is a fibered ribbon knot in $S^3=\partial B^4$ bounding a ribbon disk $D$, then given an extra transversality condition the fibration on $S^3\setminus\nu(K)$ extends to a fibration of $B^4\setminus\nu(D)$. This partially answers a question of Casson and Gordon. In particular, we show the fibration always extends when $D$ has exactly two local minima. 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 $S^1$.