Knotted handlebodies in the 4-sphere and 5-ball

TL;DR

This paper constructs specific genus-g handlebodies embedded in the 4-sphere that share boundary data but are not isotopic, confirming a conjecture for genus at least 2.

Contribution

It proves the Budney-Gabai conjecture by constructing non-isotopic handlebodies with identical boundary data in the 4-sphere for genus at least 2.

Findings

Existence of non-isotopic handlebodies with same boundary in S^4

Construction method for genus-g handlebodies in S^4

Confirmation of the Budney-Gabai conjecture for genus ≥ 2

Abstract

For every integer $g\ge 2$ we construct 3-dimensional genus-$g$ 1-handlebodies smoothly embedded in $S^4$ with the same boundary, and which are defined by the same cut systems of their boundary, yet which are not isotopic rel. boundary via any locally flat isotopy even when their interiors are pushed into $B^5$. This proves a conjecture of Budney-Gabai for genus at least 2.