Knotting and linking in 4 and 5 dimensions from barbell diffeomorphisms

TL;DR

This paper constructs explicit examples of knotted and linked structures in 4 and 5 dimensions using barbell diffeomorphisms, addressing longstanding conjectures and extending recent research.

Contribution

It provides explicit constructions of knotted 3-knots, handlebodies, and spheres in higher dimensions, resolving open problems in the field.

Findings

Constructed infinitely many non-isotopic 3-knots in the 5-sphere.

Produced knotted solid tori in 4-sphere and 5-ball.

Extended previous work with explicit examples using barbell diffeomorphisms.

Abstract

In this paper, we construct infinitely many non-isotopic 3-knots in the 5-sphere, each of which has four critical points with respect to the standard height function of the 5-sphere. This contrasts with a theorem of Scharlemann which says that any 2-knot in the 4-sphere with four critical points is unknotted, and also provides infinitely many knotted solid tori in the 4-sphere and 5-ball, which resolves the last remaining case of the conjecture by Budney and Gabai on the existence of knotted handlebodies. We also construct various knotted and linked handlebodies, discs, and spheres in the 4-sphere, 5-ball, and 5-sphere, extending recent works of Hughes, Miller, and the first author, and a recent work of the authors. All of our examples are explicit and are constructed using barbell diffeomorphisms.