\subsection{Pictures of the source and cap foams}\label{alg:foam-pictures}

We illustrate some of these basis elements before turning to the gauge
theory. The pairs $C_{14},C_{19}$ and $B_2,B_7$ each have a common
underlying undotted foam; the two members of each pair differ only in
the position of one dot. We include $B_{33}$ as well, since the expression
for $\alpha$ in \eqref{alg:scalar-tests} uses $C_{14}+C_{19}$ at its
incoming end and $\operatorname{cap}(B_{33})$ at its outgoing end.
The caps of $B_2$ and $B_7$ occur in the expression for $\chi$.

It is easiest to draw the foams in the direction of reduction, from the
web to the empty web. The creation foam is obtained by reversing this
direction. Outside the face being reduced, the edges sweep out product
sheets, or ``walls''. Within that face, sheets close off or join along
seams as in Figure~\ref{fig:elementary-foams}. There are several such
levels in a complete half-foam. In particular, a triangle reduction
has a tetrahedral point, where four seams meet. The representatives
of $C_{14},C_{19}$ have two tetrahedral points; those of $B_2,B_7$
have five, and that of $B_{33}$ has three. A single shaded subset of
the original planar web would not, by itself, specify these recorded
constructions.

\begin{figure}[p]
\centering
\resizebox{.95\textwidth}{!}{\input{figures/local_foam_models}}
\caption{Transparent views of the elementary reduction foams. Time
increases upwards. Black curves are boundary webs, blue curves are
seams, and the shaded regions are sheets. The faint vertical sides
bound the displayed portion of the product sheets. The corners may
be rounded. Each local model is extended by the product outside its
support; the terminal theta cap has no outgoing web.}
\label{fig:elementary-foams}
\end{figure}

Figures~\ref{fig:C-half-foams}--\ref{fig:B33-half-foam} give the complete
reduction sequences for these examples. In each frame the shaded region
is the planar face to which the next elementary foam is applied.
Thus it marks the support of a move, rather than an additional sheet
lying in the incoming slice. The dashed blue curves are the projections
of its seam arcs; at a triangle the fourth seam leaves the plane towards
the new vertex. The web is redrawn between frames, with its inherited
cyclic orders retained; a different face may be chosen as the unbounded
face. Vertex names are those of the reduction record;
a name may be reused after its previous vertex has disappeared.

The tuple below a frame gives the vertices of the selected face in
cyclic order. At a square, $(01)(23)$ or $(03)(12)$ specifies which
successive positions in that tuple are joined by the seam arcs.
A colored dot on an edge means a dot on the incident facet, just inside
the collar of that slice. It does not mean a marked vertex of the web.
Frame numbers start at one; edge numbers in the dot annotations retain
the zero-based convention of the supporting records. In the first two
movies the marked positions are alternatives: each of the indicated
half-foams has one dot. In the movie for $B_{33}$ both positions are used.

\clearpage
\begin{figure}[p]
\centering
\input{figures/half_foam_C}
\caption{The common reduction movie for $C_{14}$ and $C_{19}$, beginning
with $L_1$. Read the frames from left to right, then downwards. The
creation foams run through these elementary cobordisms in reverse.
The choices at the three squares are part of the specification.}
\label{fig:C-half-foams}
\end{figure}
\begin{figure}[p]
\centering
\input{figures/half_foam_B}
\caption{The common reduction movie for $B_2$ and $B_7$, beginning with
$K_1$. These are the two dotted caps whose sum defines $\ell_1$.
The five triangle steps account for the five tetrahedral points of
the undotted foam.}
\label{fig:B-half-foams}
\end{figure}
\begin{figure}[p]
\centering
\input{figures/half_foam_B33}
\caption{The reduction movie for $B_{33}$ after the first two triangle
steps of Figure~\ref{fig:B-half-foams}. The different square resolution
at step 3 changes the subsequent movie. Both indicated dots are present.
This cap represents $\ell_2$, which is used in the tests for $\alpha$
and $\beta$.}
\label{fig:B33-half-foam}
\end{figure}
\clearpage