Geometry & Topology

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

Peter Feller

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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.

Article information

Geom. Topol., Volume 20, Number 3 (2016), 1763-1771.

Received: 13 April 2015
Revised: 4 September 2015
Accepted: 6 September 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

topological slice genus Alexander polynomial


Feller, Peter. The degree of the Alexander polynomial is an upper bound for the topological slice genus. Geom. Topol. 20 (2016), no. 3, 1763--1771. doi:10.2140/gt.2016.20.1763.

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