The Dihedral Genus of a Knot

Authors

Patricia Cahn, Smith CollegeFollow
Alexandra Kjuchukova, University of Wisconsin-Madison

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

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

Rights

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

Comments

Peer reviewed accepted manuscript.

Recommended Citation

Cahn, Patricia and Kjuchukova, Alexandra, "The Dihedral Genus of a Knot" (2018). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/60