Dihedral Fox coloring counts are preserved by Reidemeister moves

For fixed $b\in {\mathbb{F}}_p$, define $R_b(a)=a\ast b=2b-a$. Then $R_b(R_b(a))=2b-(2b-a)=a$, so $R_b^{-1}=R_b$. Also $a\ast a=2a-a=a$. Thus both the forward and reverse crossing rules use the same involution, and an arc's color is uniquely recoverable across one underpass.

In an R1 disk, the coloring relation on the kinked side is either $y=x\ast x$ or $x=y\ast x$, where $x$ is the unchanged boundary color and $y$ is the color of the new interior underarc. In the first case $y=x$ by idempotency; in the second, $R_x(y)=x$ has the unique solution $y=R_x(x)=x$ because $R_x$ is bijective. Deleting the kink is the inverse restriction map. The two formulas cover both strand directions because the inverse operation equals the operation by step 1.1.

In an R2 disk, fix the color $b$ of the strand passing over the other strand and the incoming underarc color $a$. The first crossing changes the underarc color to $R_b(a)$, and the second changes it by $R_b^{-1}$; their composition is $R_b^{-1}(R_b(a))=a$. Therefore every boundary coloring extends uniquely across the two-crossing side, and the reverse R2 move gives the inverse bijection. Other orientation variants replace one of these maps by its inverse, which is the same map by step 1.1.

For a braid-like R3 orientation, start with boundary colors $a,b,c$. Propagating through the two sides gives respectively $(c,b\ast c,(a\ast b)\ast c)$ and $(c,b\ast c,(a\ast c)\ast (b\ast c))$. The last coordinates agree by right self-distributivity in [F2], so every boundary coloring on either side has a unique matching extension on the other. Reversing a crossing replaces a right translation by its inverse; step 1.1 makes that the same map, and relabeling the three strands gives every height-order and orientation variant in [F3].

If $D$ is empty, its unique empty coloring is carried to itself by planar isotopy, and no Reidemeister move is supported on the empty diagram. If $D$ is nonempty, planar isotopy carries every arc and local crossing relation bijectively to its image, while steps 2.1, 2.2, and 2.3 give explicit coloring bijections for every local move. Composing the finitely many bijections along any finite move sequence preserves the coloring count. These constructions act on finite arc labels and use no choice principle.