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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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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


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