Let be a Fox –colored knot and assume bounds a locally flat surface over which the given –coloring extends. This coloring of induces a dihedral branched cover . Its branching set is a closed surface embedded in locally flatly away from one singularity whose link is . When is homotopy ribbon and a definite four-manifold, a condition relating the signature of and the Murasugi signature of guarantees that in fact realizes the four-genus of . We exhibit an infinite family of knots with this property, each with a Fox –colored surface of minimal genus . As a consequence, we classify the signatures of manifolds which arise as dihedral covers of in the above sense.
"The dihedral genus of a knot." Algebr. Geom. Topol. 20 (4) 1939 - 1963, 2020. https://doi.org/10.2140/agt.2020.20.1939