\section*{Recycling}
This square requires some effort, and first of all a definition of the ``inflate'' operation of the bottom arrow. This operation 
is well-defined when a 1-dimensional (red) strand connects two capped tubes, as illustrated in Figure~\ref{fig:Inflate}.
Inflating replaces the 1-dimensional string with a narrow tube, creating two smooth ``pair of pants'' vertices where the string 
joined the capped strands. Now by isotopy the capped strands can be retracted and just one tube is left, as shown in Figure~\ref{fig:Inflate}. Alternatively, the 1D string can first be moved by isotopy to join near the caps, then inflating it produces
a single tube. This is well-defined. The square (7) involves the associated graded (arrow diagram) version of this operation: use the VI relation to move all arrow endings off the capped strands, then turn the thin red strand thick black, and dispose of the capped strands, as seen in square (7).

%\begin{figure}
%\input figs/Inflate.pstex_t
%\caption{Inflating a string between capped tubes.}\label{fig:Inflate}
%\end{figure}