A note on the topological sliceness of some 2-bridge knots

TL;DR

This paper demonstrates that some algebraically slice 2-bridge knots are not topologically slice using twisted Alexander polynomials, highlighting limitations of Casson-Gordon signatures in detecting sliceness.

Contribution

It introduces a novel application of twisted Alexander polynomials to distinguish topological sliceness in 2-bridge knots beyond Casson-Gordon signatures.

Findings

Twisted Alexander polynomials can obstruct topological sliceness.

Casson-Gordon signatures vanish for these knots, showing their limitations.

Computations suggest twisted Alexander polynomials are effective in this context.

Abstract

We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson-Gordon signatures in obstructing the smooth sliceness of 2-bridge knots.