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

· 2015 · math.GT · arXiv 1504.01064

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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.

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  • A note on the topological slice genus of satellite knots math.GT · 2019-08-10 · conditional · none · ref 6 · internal anchor

    The Z-slice genus of a satellite knot is at most the Z-slice genus of the pattern applied to the unknot plus the Z-slice genus of the companion knot.