Extending fibrations of 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 $S^1$.