\begin{figure}[tp]
\centering
\begin{tikzpicture}[x=0.52cm, y=0.52cm, font=\scriptsize]
\newcommand*{\plusface}[2]{\fill[blue!22] (#1,#2) rectangle ++(1,1); \node at (#1+0.5,#2+0.5) {$+$};}
\newcommand*{\minusface}[2]{\fill[red!22] (#1,#2) rectangle ++(1,1); \node at (#1+0.5,#2+0.5) {$-$};}
\newcommand*{\panelgrid}[1]{\draw[black!30, line width=0.3pt, shift={(#1,0)}, step=1] (0,0) grid (8,6);}
% panel (a)
\foreach \c/\r in {1/4,2/4,1/3,2/3,7/3,3/2,7/2,1/1,2/1,1/0,2/0}{\plusface{\c}{\r}}
\foreach \c/\r in {4/4,5/4,4/3,5/3,5/1,6/1,5/0,6/0}{\minusface{\c}{\r}}
\panelgrid{0}
\foreach \c/\r in {1/4,2/4,1/3,2/3,3/2}{\draw[line width=1.1pt] (\c,\r) rectangle ++(1,1);}
\draw[line width=1.6pt] (2.72,3.28) -- (3.28,2.72);
\draw[line width=0.8pt, dash pattern=on 1.8pt off 1.4pt] (3.28,2.28) -- (2.72,1.72);
\node at (4,-0.7) {(a) level $\ell$ of $\bar\omega$};
\node at (2.0,5.5) {$A$};
% arrow between panels
\draw[->, line width=0.8pt] (8.8,2.7) -- (10.6,2.7) node[midway, above] {$\Sigma^\ell_A$};
% panel (b)
\begin{scope}[shift={(11.4,0)}]
\foreach \c/\r in {7/3,7/2,1/1,2/1,1/0,2/0}{\plusface{\c}{\r}}
\foreach \c/\r in {1/4,2/4,1/3,2/3,3/2,4/4,5/4,4/3,5/3,5/1,6/1,5/0,6/0}{\minusface{\c}{\r}}
\draw[black!30, line width=0.3pt, step=1] (0,0) grid (8,6);
\foreach \c/\r in {1/4,2/4,1/3,2/3,3/2}{\draw[line width=1.1pt] (\c,\r) rectangle ++(1,1);}
\draw[line width=1.6pt] (2.72,3.28) -- (3.28,2.72);
\draw[line width=0.6pt, black!45, dash pattern=on 0.6pt off 1.2pt] (3.28,2.28) -- (2.72,1.72);
\draw[line width=0.8pt, dash pattern=on 1.8pt off 1.4pt] (3.72,2.72) -- (4.28,3.28);
\node at (4,-0.7) {(b) after the swap};
\end{scope}
\end{tikzpicture}
\caption{FK clusters at level $\ell$ for a pair of height functions with difference $D$. Faces with
$D>\ell$ are marked $+$, faces with $D<\ell$ are marked $-$, and blank faces have $D=\ell$. Two faces of
a diagonal across which $D-\ell$ has the same sign, while $D=\ell$ on the other diagonal, form a
\emph{saddle}. Separating the two faces of a saddle costs a factor $q=c^{-2}$ in the joint weight, paid
by lowering the residual energy of the diagonal by one; so a saddle with residual energy $0$ cannot be
separated and is an \emph{open bridge} (thick line), and otherwise it is \emph{closed} (dashed). In (a)
the outlined set $A$ is an FK cluster: four faces at the top left and the face reached through the open
bridge; the closed saddle below does not join it to the cluster there. In (b) the two copies are
exchanged on $A$, which reverses the sign of $D-\ell$ there. The open bridge inside $A$ is unaffected.
The closed saddle of (a) has become a diagonal across which $D-\ell$ changes sign (dotted), and its
residual energy was lowered by one. Conversely, the diagonal between $A$ and the minus block, across
which $D-\ell$ changed sign in (a), has become a saddle whose residual energy was raised by one, so it is
closed (dashed, right). The joint weight is unchanged (Lemma~\ref{lem:FKswap}).}
\label{fig:swap}
\end{figure}