Slice Implies Mutant Ribbon for Odd, 5-Stranded Pretzel Knots

TL;DR

This paper proves that all slice, odd, 5-stranded pretzel knots are mutant ribbon, supporting a weaker version of the Slice-Ribbon Conjecture through analysis of their concordance properties.

Contribution

It establishes that slice, odd, 5-stranded pretzel knots are mutant ribbon, advancing understanding of the Slice-Ribbon Conjecture for this family.

Findings

Certain pretzel knots have infinite order in the concordance group.

Some odd, 5-stranded pretzel knots are proven not to be slice.

The paper identifies conditions under which these knots are not slice.

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 $mutant$ $ribbon$. 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.