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The degree of the Alexander polynomial is an upper bound for the topological slice genus. (English) Zbl 1386.57008

Summary: 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.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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