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

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.

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topological slice genus, Alexander polynomial
Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Received: 13 April 2015
Revised: 4 September 2015
Accepted: 6 September 2015
Published: 4 July 2016
Proposed: Dmitri Burago
Seconded: Ciprian Manolescu, Ronald Stern
Peter Feller
Department of Mathematics
Boston College
Maloney Hall
Chestnut Hill, MA 02467
United States