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Mathematics, Geometric Topology
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 $X\to S^4$. 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.
