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The dihedral genus of a knot

Patricia Cahn and Alexandra Kjuchukova

Algebraic & Geometric Topology 20 (2020) 1939–1963

Let K S3 be a Fox p–colored knot and assume K bounds a locally flat surface S B4 over which the given p–coloring extends. This coloring of S induces a dihedral branched cover X S4. Its branching set is a closed surface embedded in S4 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 Km 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 S4 in the above sense.

knot, branched cover, ribbon genus, trisection
Mathematical Subject Classification 2010
Primary: 57M12, 57M25, 57Q60
Received: 29 December 2018
Revised: 24 July 2019
Accepted: 25 October 2019
Published: 20 July 2020
Patricia Cahn
Department of Mathematics and Statistics
Smith College
Northampton, MA
United States
Alexandra Kjuchukova
Max Planck Institute for Mathematics