Stabilization distance between surfaces

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

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 mm we find a pair of 2-knots in the 4-sphere whose stabilization distance equals mm. Next, using a generalized stabilization distance that counts connected sum with arbitrary 2-knots as distance zero, for every nonnegative integer mm we exhibit a knot J_m in the 3-sphere with two slice discs in the 4-ball whose generalized stabilization distance equals~mm. We show this using homology of cyclic covers. Finally, we use metabelian twisted homology to show that for each~mm there exists a knot and pair of slice discs with generalized stabilization distance at least mm, 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Powell, Dr Mark
• Authors: Miller, A. N., and Powell, M.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: L’Enseignement Mathématique
• Journal Abbr.: Enseign. Math.
• Publisher: EMS Press
• ISSN: 0013-8584
• ISSN (Online): 2309-4672
• Published Online: 11 June 2020
• Copyright Holders: Copyright © 2020 EMS Press
• First Published: First published in L’Enseignement Mathématique 65(3/4): 397-440
• Publisher Policy: Reproduced in accordance with the publisher copyright policy