Extending fibrations of knot complements to ribbon disk complements

Miller, Maggie

doi:10.2140/gt.2021.25.1479

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Extending fibrations of knot complements to ribbon disk complements

Maggie Miller

Geometry & Topology 25 (2021) 1479–1550

DOI: 10.2140/gt.2021.25.1479

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 $S^{3} ∖ ν (K)$ extends to a fibration of $B^{4} ∖ ν (D)$ . 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 $S^{1}$ .

Keywords

fibered, ribbon, knot, slice, topology, 4-manifold

Mathematical Subject Classification

Primary: 57K45, 57K99

Secondary: 57K10, 57K40

References

Publication

Received: 28 February 2019

Revised: 3 June 2020

Accepted: 21 July 2020

Published: 20 May 2021

Proposed: Cameron Gordon

Seconded: Mladen Bestvina, Ciprian Manolescu

Authors

Maggie Miller
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States
https://math.mit.edu/~maggiehm/

Volume 25, issue 3 (2021)