Stabilization distance between surfaces

Miller, Allison N.; Powell, Mark

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
• Public URL: https://durham-repository.worktribe.com/output/1275146
• Related Public URLs: https://arxiv.org/abs/1908.06701