The degree of the Alexander polynomial is an upper bound for the topological slice genus

TL;DR

This paper proves that the degree of the Alexander polynomial provides an upper bound for twice the topological slice genus of a knot, extending previous results and offering concrete examples where this bound is exact.

Contribution

It generalizes the relationship between Alexander polynomial degree and topological slice genus, utilizing Freedman's disc theorem to establish the bound.

Findings

The Alexander polynomial degree bounds twice the topological slice genus.

Examples where the bound precisely determines the topological slice genus.

Extension of classical knot theory results to the topological category.

Abstract

We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples of knots where this determines the topological slice genus.