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

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.

7 pages, 1 figure. Comments welcome! Version 2: Change in convention for the degree of the Alexander polynomial. Accepted for publication by Geometry and Topology

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