Document Type

Article

Publication Date

2018

Publication Title

Algebraic and Geometric Topology

Abstract

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.

Creative Commons License

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

Rights

Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy.

Comments

Peer reviewed accepted manuscript.

Included in

Mathematics Commons

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