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 $mutant$ $ribbon$. 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.

31 pages, 9 figures. Version 2 includes results for pretzel knots with single-twists (strands with twisting parameter 1 or -1), which were not addressed in version 1. Version 2 also includes one new figure and fixes a few typos

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