For the Fox n-coloring (knot theory) there is the property that
Around a crossing, the average of the colors of the undercrossing arcs equals the color of the overcrossing arc.
This matches with my idea what a coloring should do, but I can't see a deeper meaning or an mathmatical explanation/reason for this property.
2026-03-25 13:05:05.1774443905
Average property for the Fox n-coloring (knot theory)
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The deeper meaning is that, at least for prime $n$, a Fox $n$-coloring is a homomorphism $\pi_1(S^3 \setminus K) \rightarrow D_n$ mapping meridians to reflections, where $D_n$ is the dihedral group. To see this, take a rotation $r \in D_n$ of order $n$ and any reflection $s$. Then you interpret the color $k \in \mathbb Z/n$ by the reflection $r^ks \in D_n$. This defines the corresponding homomorphism since at each crossing the Wirtinger relations will be satisfied by the average condition. Of course, there are still some computations to be done and pictures to be drawn, so let me know if you need more guidance.