Slice implies mutant ribbon for odd $5$–stranded pretzel knots

Abstract

A pretzel knot $K$ is called odd if all its twist parameters are odd and mutant ribbon if it is mutant to a simple ribbon knot. We prove that the family of odd $5$–stranded pretzel knots satisfies a weaker version of the slice-ribbon conjecture: all slice odd $5$–stranded pretzel knots are mutant ribbon, meaning they are mutant to a ribbon knot. We do this in stages by first showing that $5$–stranded pretzel knots having twist parameters with all the same sign or with exactly one parameter of a different sign have infinite order in the topological knot concordance group and thus in the smooth knot concordance group as well. Next, we show that any odd $5$–stranded pretzel knot with zero pairs or with exactly one pair of canceling twist parameters is not slice.