Unknotting numbers of 2-spheres in the 4-sphere

TL;DR

This paper investigates two different measures of how complex 2-spheres are in the 4-sphere, comparing their relationships, providing bounds, examples, and applications to classical knot theory.

Contribution

It establishes bounds between the stabilization and Casson-Whitney numbers, and provides explicit examples illustrating their equality and disparity.

Findings

Stabilization number is at most one more than the Casson-Whitney number.

Explicit families of spheres where the invariants are equal or distinct.

Examples showing non-additivity of the invariants.

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.