#### Volume 20, issue 4 (2020)

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The dihedral genus of a knot

### Patricia Cahn and Alexandra Kjuchukova

Algebraic & Geometric Topology 20 (2020) 1939–1963
##### Abstract

Let $K\subset {S}^{3}$ be a Fox $p$–colored knot and assume $K$ bounds a locally flat surface $S\subset {B}^{4}$ over which the given $p$–coloring extends. This coloring of $S$ induces a dihedral branched cover $X\to {S}^{4}\phantom{\rule{-0.17em}{0ex}}$. Its branching set is a closed surface embedded in ${S}^{4}$ 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 ${K}_{m}$ with this property, each with a Fox $3$–colored surface of minimal genus $m$. As a consequence, we classify the signatures of manifolds $X$ which arise as dihedral covers of ${S}^{4}$ in the above sense.

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