Mathematical Research Letters

Volume 28 (2021)

Number 4

Pretzel links, mutation, and the slice-ribbon conjecture

Pages: 945 – 966



Paolo Aceto (Mathematical Institute, University of Oxford, United Kingdom)

Min Hoon Kim (Department of Mathematics, Chonnam National University, Gwangju, South Korea)

Junghwan Park (Department of Mathematical Sciences, KAIST, Daejeon, South Korea)

Arunima Ray (Max Planck Institut für Mathematik, Bonn, Germany)


Let $p$ and $q$ be distinct integers greater than one. We show that the $2$-component pretzel link $P(p, q, -p, -q)$ is not slice, even though it has a ribbon mutant, by using $3$-fold branched covers and an obstruction based on Donaldson’s diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for $4$-stranded $2$-component pretzel links.

Received 11 September 2019

Accepted 5 July 2020

Published 22 November 2021