Unknotting numbers of 2-spheres in the 4-sphere
Abstract
We compare two naturally arising notions of unknotting number for 2-spheres in the 4-sphere: namely, the minimal number of 1-handle stabilizations needed to obtain an unknotted surface, and the minimal number of Whitney moves required in a regular homotopy to the unknotted 2-sphere. We refer to these invariants as the stabilization number and the Casson-Whitney number of the sphere, respectively. Using both algebraic and geometric techniques, we show that the stabilization number is bounded above by one more than the Casson-Whitney number. We also provide explicit families of spheres for which these invariants are equal, as well as families for which they are distinct. Furthermore, we give additional bounds for both invariants, concrete examples of their non-additivity, and applications to classical unknotting number of 1-knots.
Keywords
Cite
@article{arxiv.2007.13244,
title = {Unknotting numbers of 2-spheres in the 4-sphere},
author = {Jason Joseph and Michael Klug and Benjamin Ruppik and Hannah Schwartz},
journal= {arXiv preprint arXiv:2007.13244},
year = {2021}
}
Comments
29 pages, 22 figures; v2 is the final draft which has been accepted for publication in Journal of Topology; v2 includes improvements to the exposition, the numbering of the theorems in the introduction and in some of the subsequent sections has changed