Stabilization distance between surfaces

Miller, Allison N.; Powell, Mark

doi:10.4171/lem/65-3/4-4

Stabilization distance between surfaces

Miller, Allison N.; Powell, Mark

Authors

Allison N. Miller

Mark Powell

Abstract

Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every nonnegative integer m we find a pair of 2-knots in the 4-sphere whose stabilization distance equals m. Next, using a generalized stabilization distance that counts connected sum with arbitrary 2-knots as distance zero, for every nonnegative integer m we exhibit a knot Jm in the 3-sphere with two slice discs in the 4-ball whose generalized stabilization distance equals~m. We show this using homology of cyclic covers. Finally, we use metabelian twisted homology to show that for each~m there exists a knot and pair of slice discs with generalized stabilization distance at least m, with the additional property that abelian invariants associated to cyclic covering spaces coincide. This detects different choices of slicing discs corresponding to a fixed metabolising link on a Seifert surface.

Citation

Miller, A. N., & Powell, M. (2019). Stabilization distance between surfaces. L’Enseignement mathématique, 65(3/4), 397-440. https://doi.org/10.4171/lem/65-3/4-4

Journal Article Type	Article
Acceptance Date	Dec 10, 2019
Online Publication Date	Jun 11, 2020
Publication Date	2019
Deposit Date	Dec 16, 2019
Publicly Available Date	Jun 11, 2021
Journal	L'Enseignement mathématique.
Print ISSN	0013-8584
Electronic ISSN	2309-4672
Publisher	EMS Press
Peer Reviewed	Peer Reviewed
Volume	65
Issue	3/4
Pages	397-440
DOI	https://doi.org/10.4171/lem/65-3/4-4
Related Public URLs	https://arxiv.org/abs/1908.06701