\section{The diagrams of six bodies}\label{sec:diagrams}

A \emph{labelled diagram} on six bodies assigns to each pair one of the kinds none, $z$, $w$, $zw$, and to each body a $z$-circle or not and a $w$-circle or not. The symmetric group $S_6$ acts by relabelling, and the exchange of the colours acts by exchanging the kinds $z$ and $w$ and the two sets of circles. By Lemma~\ref{lem:symm}, the diagram of the image of a singular place under a relabelling or under $\iota$ is the image of its diagram, so the rules of Lemma~\ref{lem:rules} are invariant under both actions. From now on we mark with $\dagger$ the statements whose own proof includes a computation. Statements that only cite them are not marked.

\begin{proposition}\cert\label{prop:enum}
Exactly $82{,}712$ labelled diagrams on six bodies satisfy the combinatorial parts of the rules \textup{(E1)--(E6)} and \textup{(C1)--(C4)} of Lemma~\ref{lem:rules} in both colours. Up to relabelling and exchange of the colours they form $191$ classes.
\end{proposition}

Here the combinatorial parts are the rules themselves, except that (E3) and (E6) are reduced to their statements on the kinds of the edges, since a diagram does not record $\zeta$ and $\omega$.

\begin{proof}
A depth-first search assigns kinds to the fifteen pairs. It checks (E2), (E3) and (E6), as conditions on the kinds, as soon as a triangle or a cycle of four pairs is complete. These checks leave $932{,}677$ assignments. At the leaves it checks (E1), (E4) and (E5), and then enumerates the circles of each colour allowed by (C1) and tests (C2)--(C4). Every labelled diagram that satisfies the rules is a leaf of the search. The classes are the orbits of the $1440$ symmetries.
\end{proof}

The program passes three checks. For four bodies it finds $41$ labelled diagrams in $5$ classes, the five four-body diagrams of \cite[\S4, Fig.~8]{AK12}. For five bodies it finds $1376$ labelled diagrams in $24$ classes. This is the count of the search checked in the Lean kernel in \cite[\S12.4]{Tom26}, and its classes are the sixteen diagrams of \cite[Fig.~11]{AK12} and eight others. For six bodies, the $117$ diagrams of Chang and Chen \cite[Figs.~3--6]{CC23} satisfy our rules and lie in $117$ distinct classes of our $191$. Each of the other $74$ classes violates one of the further rules of \cite[\S3]{CC23} or the following consequence of (E3) and (O6): a triangle of strokes of one colour does not have exactly one circled vertex of that colour. Indeed, if $k$ is the only $z$-circled vertex of a triangle $klj$ of $z$-strokes, then $\zeta_{kl}=-2$ by (O6), so $\zeta_{lj}=-2$ by (E3), contradicting $\zeta_{lj}\ge\min(a_l,a_j)>-2$. Eight of the $74$ classes violate only this rule. We do not need these rules, because Section~\ref{sec:cells} treats all $191$ classes.

We refer to a diagram by its number in \cite[Figs.~3--6]{CC23} when it has one, with the labelling of the bodies used there. Table~\ref{tab:ten} and Figure~\ref{fig:ten} show the ten diagrams of Sections~\ref{sec:ak50} and~\ref{sec:k4}. None of them has a $zw$-edge.

\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}rllll@{}}
\toprule
diagram & $z$-strokes & $z$-circles & $w$-strokes & $w$-circles\\
\midrule
64 & all pairs of $\{3,4,5,6\}$ & none & $12$ & $1,2$\\
66 & all pairs of $\{3,4,5,6\}$ & none & $12$, $16$, $26$ & $1,2$\\
81 & $56$ & $5,6$ & $34$ & $3,4$\\
82 & $56$ & $5,6$ & $23$, $24$, $34$ & $3,4$\\
88 & $56$ & $5,6$ & all pairs of $\{1,2,3,4\}$ & $1,2,3,4$\\
89 & $45$, $46$, $56$ & $5,6$ & $23$, $24$, $34$ & $2,3$\\
90 & $45$, $46$, $56$ & $5,6$ & $12$, $13$, $23$ & $2,3$\\
93 & $45$, $46$, $56$ & $5,6$ & $12$, $13$, $23$ & $1,2,3$\\
94 & $45$, $46$, $56$ & $5,6$ & all pairs of $\{1,2,3,4\}$ & $1,2,3$\\
98 & $45$, $46$, $56$ & $4,5,6$ & $12$, $13$, $23$ & $1,2,3$\\
\bottomrule
\end{tabular}
\caption{The diagrams treated in Sections~\ref{sec:ak50} and~\ref{sec:k4}, numbered and labelled as in \cite[Figs.~5, 6]{CC23}.}\label{tab:ten}
\end{table}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=0.72]
\newcommand{\dg}[6]{%
  \begin{scope}[shift={#1}]
    \foreach \k in {1,...,6} {\coordinate (v\k) at ({60*(\k-1)}:1);}
    \foreach \a/\b in {#2} {\draw[thick] (v\a)--(v\b);}
    \foreach \a/\b in {#3} {\draw[thick,dashed] (v\a)--(v\b);}
    \foreach \k in {#4} {\draw (v\k) circle (0.17);}
    \foreach \k in {#5} {\draw[densely dotted,thick] (v\k) circle (0.29);}
    \foreach \k in {1,...,6} {\fill (v\k) circle (0.05); \node[font=\tiny] at ({60*(\k-1)}:1.55) {\k};}
    \node[font=\small] at (0,-1.95) {#6};
  \end{scope}}
\dg{(0,0)}{3/4,3/5,3/6,4/5,4/6,5/6}{1/2}{}{1,2}{64}
\dg{(4.3,0)}{3/4,3/5,3/6,4/5,4/6,5/6}{1/2,1/6,2/6}{}{1,2}{66}
\dg{(8.6,0)}{5/6}{3/4}{5,6}{3,4}{81}
\dg{(12.9,0)}{5/6}{2/3,2/4,3/4}{5,6}{3,4}{82}
\dg{(17.2,0)}{5/6}{1/2,1/3,1/4,2/3,2/4,3/4}{5,6}{1,2,3,4}{88}
\dg{(0,-4.5)}{4/5,4/6,5/6}{2/3,2/4,3/4}{5,6}{2,3}{89}
\dg{(4.3,-4.5)}{4/5,4/6,5/6}{1/2,1/3,2/3}{5,6}{2,3}{90}
\dg{(8.6,-4.5)}{4/5,4/6,5/6}{1/2,1/3,2/3}{5,6}{1,2,3}{93}
\dg{(12.9,-4.5)}{4/5,4/6,5/6}{1/2,1/3,1/4,2/3,2/4,3/4}{5,6}{1,2,3}{94}
\dg{(17.2,-4.5)}{4/5,4/6,5/6}{1/2,1/3,2/3}{4,5,6}{1,2,3}{98}
\end{tikzpicture}
\caption{The diagrams of Table~\ref{tab:ten}: $z$-strokes solid, $w$-strokes dashed, $z$-circles solid, $w$-circles dotted.}\label{fig:ten}
\end{figure}