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 $S^1$.

61 pages, 52 figures, one flipbook. Version 2 agrees with published version; in particular many typos corrected and figures improved

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• Trisecting a 4-dimensional book into three chapters