The dihedral genus of a knot

Cahn, Patricia; Kjuchukova, Alexandra

doi:10.2140/agt.2020.20.1939

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The dihedral genus of a knot

Patricia Cahn and Alexandra Kjuchukova

Algebraic & Geometric Topology 20 (2020) 1939–1963

DOI: 10.2140/agt.2020.20.1939

Abstract

Let $K \subset S^{3}$ be a Fox $p$ –colored knot and assume $K$ bounds a locally flat surface $S \subset B^{4}$ over which the given $p$ –coloring extends. This coloring of $S$ induces a dihedral branched cover $X \to S^{4}$ . Its branching set is a closed surface embedded in $S^{4}$ locally flatly away from one singularity whose link is $K$ . When $S$ is homotopy ribbon and $X$ a definite four-manifold, a condition relating the signature of $X$ and the Murasugi signature of $K$ guarantees that $S$ in fact realizes the four-genus of $K$ . We exhibit an infinite family of knots $K_{m}$ with this property, each with a Fox $3$ –colored surface of minimal genus $m$ . As a consequence, we classify the signatures of manifolds $X$ which arise as dihedral covers of $S^{4}$ in the above sense.

Keywords

knot, branched cover, ribbon genus, trisection

Mathematical Subject Classification 2010

Primary: 57M12, 57M25, 57Q60

References

Publication

Received: 29 December 2018

Revised: 24 July 2019

Accepted: 25 October 2019

Published: 20 July 2020

Authors

Patricia Cahn
Department of Mathematics and Statistics Smith College Northampton, MA United States

Alexandra Kjuchukova
Max Planck Institute for Mathematics Bonn Germany

Volume 20, issue 4 (2020)