Fibered knots with the same $0$-surgery and the slice-ribbon conjecture

TL;DR

This paper explores the relationship between fibered knots, the slice-ribbon conjecture, and the Akbulut-Kirby conjecture, showing that if the slice-ribbon conjecture holds, then a modified version of the Akbulut-Kirby conjecture is false, and proposing potential counterexamples.

Contribution

It demonstrates that the slice-ribbon conjecture implies the falsity of a modified Akbulut-Kirby conjecture and provides potential fibered counterexamples to the slice-ribbon conjecture.

Findings

If the slice-ribbon conjecture is true, then the modified Akbulut-Kirby conjecture is false.

Identifies potential fibered counterexamples to the slice-ribbon conjecture.

Links the validity of the slice-ribbon conjecture to the falsity of a modified Akbulut-Kirby conjecture.

Abstract

Akbulut and Kirby conjectured that two knots with the same $0$-surgery are concordant. In this paper, we prove that if the slice-ribbon conjecture is true, then the modified Akbulut-Kirby's conjecture is false. We also give a fibered potential counterexample to the slice-ribbon conjecture.