Distinguishing mutant pretzel knots in concordance

TL;DR

This paper demonstrates that certain pretzel knots are not topologically slice despite their mutants being ribbon, using twisted Alexander polynomial obstructions.

Contribution

It introduces new sliceness obstructions for pretzel knots, distinguishing non-slice knots from their mutants with a novel application of twisted Alexander polynomials.

Findings

Many pretzel knots of specified form are not topologically slice.

Their mutants are shown to be ribbon, yet the original knots are not slice.

The method employs twisted Alexander polynomial obstructions related to cyclic covers.

Abstract

We prove that many pretzel knots of the form $P(2n,m,-2n\pm1,-m)$ are not topologically slice, even though their positive mutants $P(2n, -2n\pm1, m, -m)$ are ribbon. We use the sliceness obstruction of Kirk and Livingston related to the twisted Alexander polynomials associated to prime power cyclic covers of knots.