\draftcut
\section{Odds and Ends}\label{sec:odds}


\subsection{Motivation for circuit algebras: electronic circuits}
\label{subsec:CAMotivation}
Electronic circuits are made of ``components'' that can
be wired together in many ways. On a logical level, we only care to
know which pin of which component is connected with which other pin of
the same or other component. On a logical level, we don't really need
to know how the wires between those pins are embedded in space (see
Figures~\ref{fig:FlipFlop} and~\ref{fig:Circuit}). ``Printed Circuit
Boards'' (PCBs) are operators that make smaller components (``chips'')
into bigger ones (``circuits'') --- logically speaking, a PCB is simply a
set of ``wiring instructions'', telling us which pins on which components
are made to connect (and again, we never care precisely how the wires
are routed provided they reach their intended destinations, and ever
since the invention of multi-layered PCBs, all conceivable topologies for
wiring are actually realizable). PCBs can be composed (think ``plugging
a graphics card onto a motherboard''); the result of a composition of
PCBs, logically speaking, is simply a larger PCB which takes a larger
number of components as inputs and outputs a larger circuit. Finally,
it doesn't matter if several PCB are connected together and then the
chips are placed on them, or if the chips are placed first and the PCBs
are connected later; the resulting overall circuit remains the same.

\begin{figure}[h!]
\parpic[r]{\hspace{-1cm}\raisebox{-18mm}{$\pstex{FlipFlop}$}}
\caption{
  The J-K flip flop, a very basic memory cell, is an electronic circuit
  that can be realized using 9 components --- two triple-input ``and''
  gates, two standard ``nor'' gates, and 5 ``junctions'' in which 3 wires
  connect (many engineers would not consider the junctions to be real
  components, but we do). Note that the ``crossing'' in the middle of
  the figure is merely a projection artifact and does not indicate an
  electrical connection, and that electronically speaking, we need not
  specify how this crossing may be implemented in $\bbR^3$. The J-K flip
  flop has 5 external connections (labelled J, K, CP, Q, and Q') and hence
  in the circuit algebra of computer parts, it lives in $C_5$. In the
  directed circuit algebra of computer parts it would be in $C_{3,2}$
  as it has 3 incoming wires (J, CP, and K) and two outgoing wires
  (Q and Q').
} \label{fig:FlipFlop}
\end{figure}

\begin{figure}[h!]
\parpic[r]{\raisebox{-28mm}{\includegraphics[width=50mm]{figs/Circuit.ps}}}
\caption{
  The circuit algebra product of 4 big black components and 1 small black
  component carried out using a green wiring diagram, is an even bigger
  component that has many golden connections (at bottom). When plugged into
  a yet bigger circuit, the CPU board of a laptop, our circuit functions
  as 4,294,967,296 binary memory cells.
} \label{fig:Circuit}
\end{figure}

\subsection{Proof of Proposition \ref{prop:sKTGgens}}
\label{subsec:sKTGgensProof}

We are going to ignore strand orientations throughout this proof for 
simplicity. This is not an issue as orientation switches are allowed in
$\sKTG$ without restriction. We are also going to omit vertex signs from
the pictures given the pictorial convention stated in Section \ref{subsec:KTG}.

We need to prove that any $\sKTG$ (call it $G$) can be built from the
generators listed in the statement of the proposition, using $\sKTG$ operations. To show this, consider a Morse 
drawing of $G$, that is, a planar projection of $G$ with a height function
so that all singularities along the strands are Morse and so that every
``feature'' of the projection (local minima and maxima, crossings and
vertices) occurs at a different height.

The idea in short is to decompose $G$ into levels of this Morse drawing where at each level only one ``feature'' occurs. The levels themselves
are not $\sKTG$'s, but we show that the composition of the levels can be achieved by composing their ``closed-up'' $\sKTG$ versions followed
by some unzips. Each feature gives rise to a generator by ``closing up'' extra ends at its top and bottom. We then show that we can
construct each level using the generators and the tangle insert operation.

So let us decompose $G$ into a composition of trivalent tangles (``levels''), each of which has one ``feature'' and (possibly) some 
straight vertical strands. Note that by isotopy we can make sure that every level has strands ending at both its bottom and top,
except for the first or the last level in the case of 1-tangles. An example of level decomposition is shown in the figure below. Note that the levels are generally not
elements of $\sKTG$ (have too many ends). However, we can turn each of them into a $(1,1)$-tangle (or a 1-tangle in case of the aforementioned
top first or last levels) by ``closing up'' their tops and bottoms
by arbitrary trees. In the example below we show this for one level of the Morse-drawn $\sKTG$ containing a crossing and two vertical strands.
\begin{center}
\input{figs/MorseTangle.pstex_t}
\end{center}

Now we can compose the $\sKTG$'s obtained from closing up each level. Each tree
that we used to close up the tops and bottoms of levels determines a ``parenthesization'' of the strand endings. If these parenthesizations
match on the top of each level with the bottom of the next, then we can recreate tangle composition of the levels by composing their closed
versions followed by a number of unzips performed on the connecting trees. This is illustrated in the example below, for two consecutive levels
of the $\sKTG$ of the previous example.
\begin{center}
 \input{figs/CombineLevels.pstex_t}
\end{center}

If the trees used to close up consecutive levels correspond to different parenthesizations, then we can use insertion of the left and right associators 
(the 5th and 6th pictures of the list of generators in the statement of the theorem) to change one parenthesization to match the other. This is 
illustrated in the figure below.
\begin{center}
 \input{figs/Reassociate.pstex_t}
\end{center}

So far we have shown that $G$ can be assembled from closed versions of the levels in its Morse drawing. The closed versions of the levels of $G$ are
simpler $\sKTG$'s, and it remains to show that these can be obtained from the generators using $\sKTG$ operations. 

\parpic[r]{\input{figs/SrtandsToBubble.pstex_t}}
Let us examine what each level
might look like. First of all, in the absence of any ``features'' a level might be a single strand, in which case it is the first generator itself. 
Two parallel strands when closed up become the ``bubble'', as shown on the right.

Now suppose that a level consists of $n$ parallel strands, and that the trees used to close it up on the top and bottom are horizontal mirror images 
of each other, as shown below (if not, then this can be achieved by associator insertions and unzips). We want to show that this $\sKTG$ can
be obtained from the generators using $\sKTG$ operations. Indeed, this can be achieved by repeatedly inserting bubbles into a bubble, as shown:
\begin{center}
\input{figs/BubbleInsertions.pstex_t}
\end{center}

A level consisting of a single crossing becomes a left or right twist
when closed up (depending on the sign of the crossing). Similarly, a
single vertex becomes a bubble. A single minimum or maximum becomes a 
noose or a balloon, respectively.

It remains to see that the $\sKTG$'s obtained when closing up simple features accompanied by more through strands can be built from the generators.
A minimum accompanied by an extra strand gives rise to the $\sKTG$ obtained by sticking a noose onto a vertical strand (similarly, a balloon for a maximum). 
In the case of all the other simple features and for minima and maxima accompanied by more strands, we inserting the already generated elements into 
nested bubbles (bubbles inserted into bubbles), as in the example shown below.
This completes the proof.
\begin{center}
 \input{figs/MoreStrands.pstex_t}
\end{center}
\qed