\draftcut
\subsection{The lower-interlaced presentation $E_l$ of
$\calA^w_{\exp}$} \label{subsec:AT}

For a finite set $S$ let $\glosm{TWl}{\TW_l}(S)$ be set-theoretically
the same as $\TW(S) = \FL(S)^S\times\CW(S)$ --- we only add the ``$l$''
subscript to emphasize that $\TW_l$ carries an algebraic structure,
and that it is different from the algebraic structure on $\TW_s$,
which we will study later. Elements of $\TW_l(S)$ are ordered pairs
$\glosm{parenl}{(\lambda;\,\omega)_l}$, where $\lambda\in\FL(S)^S$,
$\omega\in\CW(S)$, and the subscript $l$ is there only to remind us of
the context.

\begin{samepage}
Set $\glosm{ElDef}{\null}$
\[ \glosm{El}{E_l}(\lambda;\,\omega)_l\coloneqq \exp(\yellowm{l}\lambda)
      \ast\exp(\yellowm{\iota}\omega)
    \in\calA^w_{\exp}(S),
  \qquad\left(\parbox{1.8in}{\centering
    ``$E_l$'' for ``\underline{E}xponentiation after using $\underline{l}$''
  }\right)
\]
\end{samepage}
\begin{samepage}
\parpic[r]{\parbox{1.75in}{
  \centering{\input{figs/El.pstex_t}}
  \vspace{-3mm}
  \begin{figcap} \label{fig:El} $E_l(\lambda;\,\omega)_l$. \end{figcap}
  \vspace{3mm}
}}
\noindent where $l\colon\FL(S)^S=A_S\oplus\tder_S\to\calA^w(S)$
is the ``lower'' Lie embedding\footnotemarkT\
of trees into ${\calA^w(S)}$
(see~\cite[Section~\ref{2-subsec:ATSpaces}]{WKO2}), where $\iota$
is the obvious inclusion of wheels ($=\CW(S)=\attr_S$) into
${\calA^w(S)}$, and where exponentiation is taken using the
stacking product~\eqref{eq:TubeProduct} of $\calA^w(S)$. A
pictorial representation of $E_l(\lambda;\,\omega)_l$ appears on the right:
Reading from the bottom up, we see ``exponentially many'' copies of
$\lambda$ (meaning, a sum over $n$ of $n$ copies with coefficient $1/n!$).
Each $\lambda$ is a linear combination of trees with one head and many
tails, which are attached to the strands in $T$ with the head below the
tails. Each copy of $\lambda$ appears on the right as a gray 
``wizard's cap'' whose tip corresponds to the head of $\lambda$,
and is therefore tipped downward. Above $\exp(l\lambda)$ is our symbolic
representation of $\exp(\iota\omega)$.

Figure~\ref{fig:El} also explains the name ``interlaced'' for this
presentation, for in it heads and tails are interlaced along the strands
of $S$ (contrast with $E_s$ in Figure~\ref{fig:Es} and with $E_f$
in Figure~\ref{fig:Ef}).
\end{samepage}

It follows from the results
of~\cite[Section~\ref{2-subsec:ATSpaces}]{WKO2} that the map
$E_l\colon\TW_l(S)\to\calA^w_{\exp}(S)$ is a set-theoretic
bijection. Hence the operations of Definition~\ref{def:Operations}
induce corresponding operations on $\TW_l(S)$. We list these within the
(long!) definition-proposition below.

\footnotetextT{We could have equally well used the ``upper'' Lie embedding
$u$, setting $\glosm{Eu}{E_u}\glosm{parenu}{(\lambda;\,\omega)_u}
\coloneqq \exp(\iota\omega)\exp(\glosm{u}{u}\lambda)$, with only minor
modifications to the formulas that follow.}

\begin{defprop} \label{dp:ElOps}
The bijection $E_l$ intertwines the 
operations defined below with the operations in
Definition~\ref{def:Operations}:\footnoteC{We cannot verify
Definition-Proposition~\ref{dp:ElOps} {\it per se} on the computer, as we have
no direct computer implementation of $\calA^w$. Indeed, the whole point
of this paper is to provide an implementation of $\calA^w$ by means of
$E_l$ (and later, $E_s$ and $E_f$). Instead, we verify below that many
properties of operations on $\calA^w$ (the associativity of the stacking
product, etc.)  indeed hold for their $E_l$ implementations. We start
by setting the values of some ``sample'' elements on which we will run
our tests (note that on the computer we represent $(\lambda;\,\omega)_l$
as {\tt El[$\lambda$,$\omega$]}):

\dialoginclude{ElSetup}
}

\begin{enumerate}[leftmargin=*,labelindent=0pt]

\item If $S_1\cap S_2=\emptyset$ and $(\lambda_i;\,\omega_i)_l\in\TW_l(S_i)$,
$\glosm{lsqcup}{\null}$
\begin{equation} \label{eq:ElCup}
  (\lambda_1;\,\omega_1)_l(\lambda_2;\,\omega_2)_l
  = (\lambda_1;\,\omega_1)_l\yellowm{\sqcup}(\lambda_2;\,\omega_2)_l
  \coloneqq (\lambda_1\sqcup\lambda_2;\,\omega_1+\omega_2)_l,
\end{equation}
where $\sqcup\colon\FL(S_1)^{S_1}\times\FL(S_2)^{S_2}\to\FL(S_1\sqcup
S_2)^{S_1\sqcup S_2}$ is the union operation of functions (or, in computer
speak, the concatenation of associative arrays) followed by
the inclusions $\FL(S_i)\to\FL(S_1\sqcup S_2)$, and $\omega_1+\omega_2$
is defined using the inclusions $\CW(S_i)\to\CW(S_1\sqcup S_2)$.

\item If $(\lambda_i;\,\omega_i)_l\in\TW_l(S)$,$\glosm{last}{\null}$
\begin{equation} \label{eq:ElProduct}
  (\lambda_1;\,\omega_1)_l\yellowm{\ast}(\lambda_2;\,\omega_2)_l
  \coloneqq (
    \BCH_{tb}(\lambda_1,\lambda_2);\,
    e^{-\partial_{\lambda_2}}(\omega_1)+\omega_2
  )_l.\footnotemarkC
\end{equation}

\footnotetextC{We quote the $E_l$ implementation of the stacking product
from \href{\web/AwCalculus.m}{\tt AwCalculus.m}
(\href{\web/AwCalculus.m}{\tt AC}) and verify that it is associative,
at least to degree $8$:
\ACQuote{ElStackingDef}

\shortdialoginclude{ElAssociativity}
}

\item If $(\lambda;\,\omega)_l\in\TW_l(S)$ and $a\in S$,$\glosm{ldeta}{\null}$
\begin{equation} \label{eq:ElEta}
  (\lambda;\,\omega)_l\act \yellowm{d\eta^a}
  \coloneqq ((\lambda\yellowm{\remove}a)\act(a\to 0);\,\omega\act(a\to 0))_l,
\end{equation}
where $\lambda\remove a$ denotes the function $\lambda$ with the element
$a$ removed from its domain (in computer talk, ``remove the key $a$''),
and $(a\to 0)$ denotes the substitution $a=0$, which is defined on both
$\FL$ and $\CW$ and maps $\FL(S)\to\FL(S\remove a)$ and
$\CW(S)\to\CW(S\remove a)$.\footnoteC{Example:

\shortdialoginclude{detaExample}
}

\item For a single $a\in S$, I don't know a simple description
of the operation $\dA^a$ in $E_l$ language\footnoteT{\label{foot:notsimple}
  A not-so-simple description would be to use the language of the factored
  presentation of Section~\ref{subsec:Ef}, converting back and forth
  using the results of Section~\ref{subsec:Conversion}.
}.
Yet the composition
$\glosm{ldA}{\dA}\coloneqq\yellowm{\dA^S}\coloneqq\prod_{a\in S}\dA^a$
is manageable: ($j$ is defined in Definition~\ref{def:j})
\begin{equation} \label{eq:ElA}
  (\lambda;\,\omega)_l\act \dA^S \coloneqq
  (-\lambda;\,e^{\partial_\lambda}(\omega)-j(\lambda))_l.\footnotemarkC
\end{equation}

\footnotetextC{We quote the computer-definition of $\dA$, compute an
example, verify that $\dA$ is an involution, and then that it is an
anti-homomorphism relative to the stacking product:
\ACQuote{EldA}

\shortdialoginclude{dA1}

\shortdialoginclude{dA2}

\shortdialoginclude{dA3}
}

\addtocounter{footnoteT}{-1}%\addtocounter{HfootnoteT}{-1}
\item For a single $a\in S$, I don't know a simple description
of the operation $\dS^a$ in $E_l$
language\footnotemarkT. Yet the composition
$\glosm{ldS}{\dS}\coloneqq\yellowm{\dS^S}\coloneqq\prod_{a\in S}\dS^a$
is manageable:
\begin{equation} \label{eq:ElS}
  (\lambda;\,\omega)_l\act \dS^S
  \coloneqq (-\lambda\act(-1)^{\deg};\,
    (e^{\partial_\lambda}(\omega)-j(\lambda))\act(-1)^{\deg})_l.\footnotemarkC
\end{equation}

\footnotetextC{An example:

\shortdialoginclude{dS}
}

\addtocounter{footnoteT}{-1}%\addtocounter{HfootnoteT}{-1}
\item I don't know a simple description
of the operation $dm^{ab}_c$ in $E_l$ language\footnotemarkT. Yet note that
Equation~\eqref{eq:multiplem} implies that ``applying $dm$ to all strands''
is manageable, being the stacking product described
in~\eqref{eq:ElProduct}.

\item We have\glosm{ldDelta}{$\null$}
\begin{equation} \label{eq:ElDelta}
  (\lambda;\,\omega)_l\act \yellowm{d\Delta}^a_{bc} \coloneqq (
    (\lambda\remove a)\sqcup(b\to\lambda_a,\,c\to\lambda_a)
      \act (a\to b+c);\,
    \omega \act (a\to b+c)
  )_l,
\end{equation}
where $(a\to b+c)$ denotes the obvious replacement of the generator $a$
with the sum $b+c$. It represents morphisms $\FL(S)\to\FL((S\remove
a)\sqcup \{b,c\})$, $\FL(S)^H\to\FL((S\remove a)\sqcup \{b,c\})^H$ (for any
set $H$), and  $\CW(S)\to\CW((S\remove a)\sqcup \{b,c\})$.\footnoteC{The
computer-definition, an example, and then a verification that $d\Delta$
is homomorphism relative to
the stacking product:
\ACQuote{EldDelta}

\shortdialoginclude{dD1}

\shortdialoginclude{dD2}
}

\item We have\glosm{ldsigma}{$\null$}
\begin{equation} \label{eq:ElSigma}
  (\lambda;\,\omega)_l\act \yellowm{d\sigma}^a_b \coloneqq (
    ((\lambda\remove a)\sqcup(b\to\lambda_a))\act (a\to b);\,
    \omega\act (a\to b)
  )_l,
\end{equation}
where $(a\to b)$ denotes the obvious ``generator renaming''
morphisms $\FL(S)\to\FL((S\remove a)\sqcup b)$, $\FL(S)^H\to\FL((S\remove
a)\sqcup b)^H$ (for any set $H$), and $\CW(S)\to\CW((S\remove a)\sqcup b)$.

\end{enumerate}
\end{defprop}

\begin{proof} Equations \eqref{eq:ElCup}, \eqref{eq:ElEta},
\eqref{eq:ElDelta}, and \eqref{eq:ElSigma} are trivial and
were stated only to introduce notation. The tree-level
part of Equation~\eqref{eq:ElProduct} follows from the
fact that $l$ is a morphism of Lie algebras (see within the
proof of~\cite[Proposition~\ref{2-prop:Pnses}]{WKO2}). The
wheels part of Equation~\eqref{eq:ElProduct} follows
from~\cite[Remark~\ref{2-rem:tderontr}]{WKO2}. Equation~\eqref{eq:ElA}
follows from the observation that $\dA^S$ is the adjoint map
$\ast$ of~\cite[Definition~\ref{2-def:Adjoint}]{WKO2} and then
from~\cite[Proposition~\ref{2-prop:Jandj}]{WKO2}. Equation~\eqref{eq:ElS}
is the easily-established fact that on $\calA^w$,
$\dS^S=(-1)^{\deg}\dA^S$. \qed
\end{proof}

\Topology Note that the absence of simple descriptions of $dA^a$, $\dS^a$,
and $dm^{ab}_c$ in the $E_l$ language is fatal for its applicability to
knot theory, as these operations are needed within the computation of 
knot and tangle invariants. See Section~\ref{subsec:TangleInvariants}.

\refstepcounter{theorem}
\AT {\em Comment}~\thetheorem. \label{com:ElAT}
Let $\glosm{piT}{\pi_T}\colon\TW(S)\to\FL(S)^S$ denote the projection
onto the first factor (``trees'') of $\TW(S)=\FL(S)^S\times\CW(S)$,
and recall that up to a minor central factor, $(\FL(S)^S,\,tb)$
is $\tder_S$. Recall also that $\tder_S$ is the Lie algebra of
$\TAut_S$, and that elements of $\tder_S$ represent elements of
$\TAut_S$ by exponentiation. With this in mind, the tree part of
Equation~\eqref{eq:ElProduct} becomes the product of $\TAut_S$.
In other words, the diagram
\[ \xymatrix{
  \TW_l(S)\times\TW_l(S) \ar[r]^-\ast
    \ar[d]_{(\pi_T\act\exp)\times(\pi_T\act\exp)} &
  \TW_l(S) \ar[d]^{\pi_T\act\exp} \\
  \TAut_S\times\TAut_S \ar[r]^-{\text{mult.}} &
  \TAut_S
} \]
is commutative. Hence the $E_l$ presentation is valuable for~\cite{AT} as
many of the~\cite{AT} equations involve the group structure of $\TAut_S$.