Volume 20, issue 3 (2016)

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

Peter Feller

Geometry & Topology 20 (2016) 1763–1771
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.

Keywords
topological slice genus, Alexander polynomial
Mathematical Subject Classification 2010
Primary: 57M25, 57M27