\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}


The diagram on the right shows the spaces and maps in play.
The horizontal maps of the diagram  are the obvious inclusions 
 sending each generator
of $\wTF$ or $\bar{\calA}^w$ to the same generator of the corresponding extended space. These maps are injective as none of the relations of $\wTFe$ or 
$\widetilde{\calA}^{sw}$ change the skeleta. 
 
\parpic[r]{\input{figs/PunctureExample.pstex_t}}
First assume that $\bar{Z}^w$ is a homomorphic expansion for $\wTF$, we show that it extends uniquely to a homomorphic extension of $\wTFe$. 
$\bar{Z}^w$ assigns values in $\calA^{sw}$ to the crossings, vertices and cap; hence we only need to define the 
value of ${Z}^w$ on generators of $\wTFe$ which involve thin red strands. If such an 
extension exists, then it is unique: the generators of $\wTFe$ which involve thin red strands are obtained via puncture operations from generators which do not. 
Since $Z^w$ respects punctures, so must be the corresponding values.


It is straightforward to check that these values satisfy all equations arising from the relations of $\wTFe$ and ${Z}^w$ is indeed an expansion. 
Furthermore, ${Z}^w$ is homomorphic by design, using the homomorphicity of $\bar{Z}^w$.

Conversely, given a homomorphic expansion $\widetilde{Z}^w:\wTFe \to \widetilde{A}^{sw}$, then the corresponding homomorphic expansion $Z^w$ for $\wTF$ is 
the restriction (i.e., the composition of the inclusion of $\wTF$ into $\wTFe$ with ${Z}^w$).