\draftcut
\section{Odds and Ends} \label{sec:OddsAndEnds}
\draftcut
\subsection{Some Dimensions} \label{subsec:SomeDimensions}
The table below lists what we could find about $\calA^v$ and $\calA^w$ by
crude brute force computations in low degrees. We list degrees 0 through
7. The spaces we study are $\calA^-(\uparrow)$, $\calA^{s-}(\uparrow)$
(the $-$ in the subscript means ``$v$ and $w$''), and
$\calA^{r-}(\uparrow)$ which is $\calA^-(\uparrow)$ moded out by
``isolated'' arrows\footnote{That is, $\calA^{r-}(\uparrow)$
is $\calA^-(\uparrow)$ modulo ``framing independence'' (\glost{FI})
relations (see Section~\ref{subsec:Jacobi}, cf. ~\cite{Bar-Natan:OnVassiliev}, with the isolated arrow
taken with either orientation). It is the space related to finite type
invariants of unframed knots, on which the R1 move
is also imposed, in the same way as $\calA^-(\uparrow)$ is related
to framed knots.}, $\calP^-(\uparrow)$ which is the space of
primitives in $\calA^-(\uparrow)$, and $\glos{\calA^-(\bigcirc)}$,
$\glos{\calA^{s-}(\bigcirc)}$, and $\glos{\calA^{r-}(\bigcirc)}$,
which are the same as $\calA^-(\uparrow)$, $\calA^{s-}(\uparrow)$,
and $\calA^{r-}(\uparrow)$ except with closed knots (knots with
a circle skeleton) replacing long knots. Each of these spaces we
study in three variants: the ``v'' and the ``w'' variants, as well
as the \underline{u}sual knots ``u'' variant which is here just for
comparison. We also include a row ``$\dim\calG_m\calL ie^-(\uparrow)$''
for the dimensions of ``Lie-algebraic weight systems''. Those
are explained in the u and v cases in~\cite{Bar-Natan:OnVassiliev,
Haviv:DiagrammaticAnalogue, Leung:CombinatorialFormulas}, and in the w
case in Section~\ref{subsec:LieAlgebras}.
{
%\def\uvw#1#2#3{{\text{\tiny $\begin{array}{c}#1\\#2\\#3\end{array}$}}}
\def\uvw#1#2#3{{\hspace{-2.5mm}\text{\small
$\begin{array}{c}#1\mid#2\\#3\end{array}$}\hspace{-3mm}
}}
\begin{center}\begin{tabular}{||c|c||c|c|c|c|c|c|c|c|c||}
\hline \hline
&& \multicolumn{3}{c|}{\footnotesize See Section~\ref{subsec:ToTwo}} &&&&&& \\
$m$ && 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \footnotesize Comments \\
\hline
$\dim\calG_m\calA^-(\uparrow)$ & \uvw{u}{v}{w} &
\uvw{1}{1}{1} & \uvw{1}{2}{2} & \uvw{2}{7}{4} & \uvw{3}{27}{7} &
\uvw{6}{139}{12} & \uvw{10}{813}{19} & \uvw{19}{?}{30} & \uvw{33}{?}{45}
& \uvw{\ref{com:uknots}}{\ref{com:longv}}{\ref{com:wknots},
\ref{com:longw}, \ref{com:nextfew}} \\
\hline
$\dim\calG_m\calL ie^-(\uparrow)$ & \uvw{u}{v}{w} &
\uvw{1}{1}{1} & \uvw{1}{2}{2} & \uvw{2}{7}{4} & \uvw{3}{27}{7} &
\uvw{6}{\,\geq\!128}{12} & \uvw{10}{?}{19} & \uvw{19}{?}{30} &
\uvw{33}{?}{45}
& \uvw{\ref{com:uknots}}{\ref{com:Lie}}{\ref{com:nextfew}} \\
\hline
$\dim\calG_m\calA^{s-}(\uparrow)$ & \uvw{u}{v}{w} &
\uvw{-}{1}{1} & \uvw{-}{1}{1} & \uvw{-}{3}{2} & \uvw{-}{10}{3} &
\uvw{-}{52}{5} & \uvw{-}{298}{7} & \uvw{-}{?}{11} & \uvw{-}{?}{15}
& \uvw{\ref{com:su}}{\ref{com:longv}}{\ref{com:wknots}, \ref{com:nextfews}} \\
\hline
$\dim\calG_m\calA^{r-}(\uparrow)$ & \uvw{u}{v}{w} &
\uvw{1}{1}{1} & \uvw{0}{0}{0} & \uvw{1}{2}{1} & \uvw{1}{7}{1} &
\uvw{3}{42}{2} & \uvw{4}{246}{2} & \uvw{9}{?}{4} & \uvw{14}{?}{4}
& \uvw{\ref{com:uknots}}{\ref{com:fiwarning}}{\ref{com:wknots},
\ref{com:nextfewr}} \\
\hline
$\dim\calG_m\calP^-(\uparrow)$ & \uvw{u}{v}{w} &
\uvw{0}{0}{0} & \uvw{1}{2}{2} & \uvw{1}{4}{1} & \uvw{1}{15}{1} &
\uvw{2}{82}{1} & \uvw{3}{502}{1} & \uvw{5}{?}{1} & \uvw{8}{?}{1}
& \uvw{\ref{com:uknots}}{\ref{com:Pv}}{\ref{com:wknots}} \\
\hline
$\dim\calG_m\calA^-(\bigcirc)$ & \uvw{u}{v}{w} &
\uvw{1}{1}{1} & \uvw{1}{1}{1} & \uvw{2}{2}{1} & \uvw{3}{5}{1} &
\uvw{6}{19}{1} & \uvw{10}{77}{1} & \uvw{19}{?}{1} & \uvw{33}{?}{1}
& \uvw{\ref{com:uknots}}{\ref{com:closedv}}{\ref{com:wknots}} \\
\hline
$\dim\calG_m\calA^{s-}(\bigcirc)$ & \uvw{u}{v}{w} &
\uvw{-}{1}{1} & \uvw{-}{1}{1} & \uvw{-}{1}{1} & \uvw{-}{2}{1} &
\uvw{-}{6}{1} & \uvw{-}{23}{1} & \uvw{-}{?}{1} & \uvw{-}{?}{1}
& \uvw{\ref{com:su}}{\ref{com:longv}}{\ref{com:wknots}} \\
\hline
$\dim\calG_m\calA^{r-}(\bigcirc)$ & \uvw{u}{v}{w} &
\uvw{1}{1}{1} & \uvw{0}{0}{0} & \uvw{1}{0}{0} & \uvw{1}{1}{0} &
\uvw{3}{4}{0} & \uvw{4}{17}{0} & \uvw{9}{?}{0} & \uvw{14}{?}{0}
& \uvw{\ref{com:uknots}}{\ref{com:closedv}}{\ref{com:wknots}} \\
\hline \hline
\end{tabular}\end{center}
}
\begin{comments} \begin{enumerate}
\item \label{com:uknots} Much more is known computationally on the u-knots
case. See especially~\cite{Bar-Natan:OnVassiliev, Bar-Natan:Computations,
Kneissler:Twelve, Amir-KhosraviSankaran:VasCalc}.
\item \label{com:longv} These dimensions were computed by Louis Leung and
DBN using a program available at~\cite[``Dimensions'']{WKO}.
\item \label{com:wknots} As we have seen in Section~\ref{subsec:Jacobi},
the spaces associated with w-knots are understood to all degrees.
\item \label{com:longw} To degree 4, these numbers were also verified
by~\cite[``Dimensions'']{WKO}.
\item \label{com:nextfew} The next few numbers in these sequences are 67, 97,
139, 195, 272.
\item \label{com:Lie} These dimensions were computed by Louis Leung and
DBN using a program available at~\cite[``Arrow Diagrams and
$\mathfrak{gl}(N)$'']{WKO}. Note the match with the row above.
\item \label{com:su} There is no ``s'' quotient in the ``u'' case.
\item \label{com:nextfews} The next few numbers in this sequence are 22, 30,
42, 56, 77.
\item \label{com:fiwarning} These numbers were computed
by~\cite[``Dimensions'']{WKO}. Contrary to the $\calA^u$
case, $\calA^{rv}$ is {\em not} the quotient of $\calA^{v}$ by the
ideal generated by degree 1 elements, and therefore the dimensions
of the graded pieces of these two spaces cannot be deduced from each
other using the Milnor-Moore theorem.
\item \label{com:nextfewr} The next few numbers in this sequence are
7,8,12,14,21.
\item \label{com:Pv} These dimensions were deduced from the dimensions of
$\calG_m\calA^v(\uparrow)$ using the Milnor-Moore theorem.
\item \label{com:closedv} Computed
by~\cite[``Dimensions'']{WKO}. Contrary to the $\calA^u$
case, $\calA^v(\bigcirc)$, $\calA^{sv}(\bigcirc)$, and
$\calA^{rv}(\bigcirc)$ are {\em not} isomorphic to $\calA^v(\uparrow)$,
$\calA^{sv}(\uparrow)$, and $\calA^{rv}(\uparrow)$ and separate
computations are required.
\end{enumerate}
\end{comments}
\subsection{What Means ``Closed Form''?} \label{subsec:ClosedForm}
As stated earlier, one of our hopes for this sequence of papers is that it will
lead to closed-form formulae for tree-level associators. The
notion ``closed-form'' in itself requires an explanation. Is $e^x$ a closed form expression for
$\sum_{n=0}^\infty\frac{x^n}{n!}$, or is it just an artificial name given
for a transcendental function we cannot otherwise reduce? Likewise,
why not call some tree-level associator $\Phi^\text{tree}$ and now it is
``in closed form''?
For us, ``closed-form'' should mean ``useful for computations''. More
precisely, it means that the quantity in question is an element of some
space $\calAcf$ of ``useful closed-form thingies'' whose elements have
finite descriptions (hopefully, finite and short) and on which some
operations are defined by algorithms which terminate in finite time
(hopefully, finite and short). Furthermore, there should be a finite-time
algorithm to decide whether two descriptions of elements of $\calAcf$
describe the same element\footnote{In our context, if it is hard to
decide within the target space of an invariant whether two elements
are equal or not, the invariant is not too useful in deciding whether
two knotted objects are equal or not.}. It is even better if the said
decision algorithm takes the form ``bring each of the two elements in question
to a canonical form by means of some finite (and hopefully short)
procedure, and then compare the canonical forms verbatim''; if this is the
case, then many algorithms that involve managing a large number of elements
become simpler and faster.
Thus, for example, polynomials in a variable $x$ are always of closed form,
for they are simply described by finite sequences of integers (which in
themselves are finite sequences of digits), the standard operations on
polynomials ($+$, $\times$, and, say, $\frac{d}{dx}$) are algorithmically
computable, and it is easy to write the ``polynomial equality'' computer
program. Likewise for rational functions and even for rational functions
of $x$ and $e^x$.
On the other hand, general elements $\Phi$ of the space
$\calA^\text{tree}(\uparrow_3)$ of potential tree-level associators
are not closed-form, for they are determined by infinitely many
coefficients. Thus, iterative constructions of associators, such
as the one in~\cite{Bar-Natan:NAT} are computationally useful only
within bounded-degree quotients of $\calA^\text{tree}(\uparrow_3)$
and not as all-degree closed-form formulae. Likewise, ``explicit''
formulae for an associator $\Phi$ in terms of multiple $\zeta$-values
(e.g.~\cite{LeMurakami:HOMFLY}) are not useful for computations as it
is not clear how to apply tangle-theoretic operations to $\Phi$ (such as
$\Phi\mapsto\Phi^{1342}$ or $\Phi\mapsto(1\otimes\Delta\otimes 1)\Phi$)
while staying within some space of ``objects with finite description in
terms of multiple $\zeta$-values''. And even if a reasonable space of such
objects could be defined, it remains an open problem to decide whether
a given rational linear combination of multiple $\zeta$-values is equal
to $0$.
\draftcut
\subsection{Arrow Diagrams up to Degree 2} \label{subsec:ToTwo} Just as
an example, in this section we study the spaces $\calA^-(\uparrow)$,
$\calA^{s-}(\uparrow)$, $\calA^{r-}(\uparrow)$, $\calP^-(\uparrow)$,
$\calA^-(\bigcirc)$, $\calA^{s-}(\bigcirc)$, and $\calA^{r-}(\bigcirc)$ in
degrees $m\leq 2$ in detail, both in the ``v'' case and in the ``w'' case
(the ``u'' case has been known since long \cite{Bar-Natan:OnVassiliev,
Kneissler:Twelve, Bar-Natan:Computations}).
\subsubsection{Arrow Diagrams in Degree 0} There is only
one degree 0 arrow diagram, the empty diagram $D_0$ (see
Figure~\ref{fig:Deg0-2Diagrams}). There are no relations, and thus,
$\{D_0\}$ is the basis of all $\calG_0\calA^-(\uparrow)$ spaces
and its closure, the empty circle, is the basis of all
$\calG_0\calA^-(\bigcirc)$ spaces. $D_0$ is the unit $1$, yet $\Delta
D_0=D_0\otimes D_0=1\otimes 1\neq D_0\otimes 1+1\otimes D_0$, so $D_0$
is not primitive and $\dim\calG_0\calP^-(\uparrow)=0$.
\subsubsection{Arrow Diagrams in Degree 1} \label{subsubsec:DegreeOne}
There are only two degree 1 arrow diagrams, the ``right
arrow'' diagram $D_R$ and the ``left arrow'' diagram $D_L$ (see
Figure~\ref{fig:Deg0-2Diagrams}). There are no $6T$ relations, and
thus, $\{D_R, D_L\}$ is the basis of $\calG_1\calA^-(\uparrow)$. Modulo
RI, $D_L=D_R$ and hence, $D_A:=D_L=D_R$ is the single basis element of
$\calG_1\calA^{s-}(\uparrow)$. Both $D_R$ and $D_L$ vanish modulo FI, so
$\dim\calG_1\calA^{r-}(\uparrow)=\dim\calG_1\calA^{r-}(\bigcirc)=0$. Both
$D_R$ and $D_L$ are primitive, so $\dim\calG_1\calP^-(\uparrow)=2$.
Finally, the closures ${\bar D}_R$ and ${\bar D}_L$ of $D_R$ and $D_L$ are
equal, so $$\calG_1\calA^{s-}(\bigcirc)=\calG_1\calA^-(\bigcirc)=\langle
{\bar D}_R\rangle=\langle {\bar D}_L\rangle=\langle {\bar D}_A\rangle.$$
\begin{figure}
\[ \pstex{Deg0-2Diagrams} \]
\caption{The 15 arrow diagrams of degree at most 2.}
\label{fig:Deg0-2Diagrams}
\end{figure}
\subsubsection{Arrow Diagrams in Degree 2} There are 12 degree
2 arrow diagrams, which we denote $D_1,\dots,D_{12}$ (see
Figure~\ref{fig:Deg0-2Diagrams}). There are six $6T$ relations,
corresponding to the 6 ways of ordering the 3 vertical strands that
appear in a $6T$ relation (see Figure~\ref{fig:6T}) along a long line. The
ordering $(ijk)$ becomes the relation $D_3+D_9+D_3=D_6+D_3+D_6$. Likewise,
$(ikj)\mapsto D_6+D_1+D_{11}=D_3+D_5+D_1$, $(jik)\mapsto
D_{10}+D_2+D_6=D_2+D_5+D_3$, $(jki)\mapsto D_4+D_7+D_1=D_8+D_1+D_{11}$,
$(kij)\mapsto D_2+D_7+D_4=D_{10}+D_2+D_8$, and $(kji)\mapsto
D_8+D_4+D_8=D_4+D_{12}+D_4$. After some linear algebra, we find
that $\{D_1, D_2, D_6, D_8, D_9, D_{11}, D_{12}\}$ form a basis
of $\calG_2\calA^v(\uparrow)$, and that the remaining diagrams
reduce to the basis as follows: $D_3=2D_6-D_9$, $D_4=2D_8-D_{12}$,
$D_5=D_9+D_{11}-D_6$, $D_7=D_{11}+D_{12}-D_8$, and $D_{10}=D_{11}$. In
$\calG_2\calA^{sv}(\uparrow)$ we further have that $D_5=D_6$, $D_7=D_8$,
and $D_9=D_{10}=D_{11}=D_{12}$, and so $\calG_2\calA^{sv}(\uparrow)$
is 3-dimensional with basis $D_1$, $D_2$, and $D_3=\ldots=D_{12}$.
In $\calG_2\calA^{rv}(\uparrow)$ we further have that $D_{5-12}=0$.
Thus $\{D_1, D_2\}$ is a basis of $\calG_2\calA^{rv}(\uparrow)$.
There are 3 OC relations to write for $\calG_2\calA^w(\uparrow)$:
$D_2=D_{10}$, $D_3=D_6$, and $D_4=D_8$. Along with the $6T$ relations,
we find that $$\{D_1, D_3=D_6=D_9, D_2=D_5=D_7=D_{10}=D_{11},
D_4=D_8=D_{12}\}$$ is a basis of $\calG_2\calA^w(\uparrow)$. Similarly
$\{D_1,D_2=\ldots=D_{12}\}$ is a basis of the two-dimensional
$\calG_2\calA^{sw}(\uparrow)$. When we mod out by FI, only one
diagram remains non-zero in $\calG_2\calA^{rw}(\uparrow)$ and it is $D_1$.
We leave the determination of the primitives and the spaces with a circle
skeleton as an exercise to the reader.