Pretzel links, mutation, and the slice-ribbon conjecture

TL;DR

This paper demonstrates that certain pretzel links are not slice despite having ribbon mutants, using advanced topological tools, and confirms the slice-ribbon conjecture for specific 4-stranded pretzel links.

Contribution

It introduces a novel application of 3-fold branched covers and Donaldson's theorem to distinguish non-slice pretzel links with ribbon mutants.

Findings

P(p,q,-p,-q) links are not slice despite ribbon mutants.

Confirmed the slice-ribbon conjecture for 4-stranded 2-component pretzel links.

Abstract

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.