Algebraic and Geometric Topology
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 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.
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Cahn, Patricia and Kjuchukova, Alexandra, "The Dihedral Genus of a Knot" (2018). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.