# The Dihedral Genus of a Knot

## TL;DR

This paper explores the relationship between Fox p-colored knots, dihedral branched covers, and the four-genus of knots, providing classification results and conditions linking knot signatures to four-manifold invariants.

## Contribution

It introduces a new connection between dihedral covers, knot signatures, and the four-genus, including a classification of signatures of certain dihedral cover manifolds.

## Key findings

- Identifies conditions under which the four-genus is realized by a minimal genus surface.
- Classifies signatures of dihedral cover 4-manifolds arising from p-colored knots.
- Constructs infinite families of knots with prescribed properties.

## 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.