\section{Recycling} {\red This section does not exist in a respectable math paper.}
\subsection{Recycled 170704}
There is a standard ``quantum algebra'' methodology that associates
a framed knot / tangle invariant to certain triples $(U,R,C)$, where $U$ is
a unital algebra and $R\in U\otimes U$ and $C\in U$ are invertible
(see e.g.~\cite[Section~4.2]{Ohtsuki:QuantumInvariants}). In
Aside~\ref{aside:Tangles} we provisionally explain what we mean by ``tangle'',
and the ``quantum algebra'' methodology is recalled in
Aside~\ref{aside:URCMethod}.
\begin{Aside}\fbox{\begin{minipage}{0.9\linewidth}\sl
\parpic[r]{\input{figs/ProvisionalTangle.pdf_t}}
Like elsewhere, for us a ``tangle'' $K$ is a part of a (multi-component,
oriented, framed) knot in a part $P$ of a plane in which an ``up'' direction is
declared. Unlike elsewhere, we do not insist that $P$ would be a disk;
it may be a union of disks with a few sub-disks removed. We do insist,
however, that the ends of $K$ would lie within the boundary $\partial P$
of $P$ and would be up-going there. We also insist that the components
(``strands'') of $K$ would be intervals (i.e., not circles), and that
they would be placed in a bijection with some finite set $S$ of ``strand
labels''.
In Section~\ref{sec:RVT} we replace this provisional definition with ``rotational
virtual tangles'' in the spirit of~\cite{Kauffman:RotationalVirtualKnots}.
\caption{Provisionally, what we mean by a ``tangle''.} \label{aside:Tangles}
\end{minipage}
}\end{Aside}
\subsubsection{Formulas and Meta-Algebras} Our approach to the computation
of $z(K)$ is different. Instead of working directly in $U^{\otimes
S}$, our invariant $Z(K)$ takes values in spaces $\calF(S)$ of
``formulas for elements of $U^{\otimes S}$'' that have an ``value map''
$\bbV\colon\calF(S)\to U^{\otimes S}$, taking a formula in $\calF(S)$
to its value in $U^{\otimes S}$, for which $z=Z\act\bbV$.\footnoteT{We
use properly ordered compositions! $f\act g$ means ``do $f$ then $g$'',
often obfuscated using ``$g\circ f$''.} We make sure that the following
five properties hold:
\begin{enumerate}
\item There are simple and easy to compute (constant time) formulas for the invariants
of a crossing and of a cuap.
\item There are operations on $\calF(S)$ that mirror standard operations
on the space $U^{\otimes S}$ and on the space $\calK(S)$ of $S$-component
tangles, so that a diagram of the following nature commutes:
\[ \xymatrix@C=0.3in@R=0.2in{
\left\{\calK(S)\right\}
\ar@`{p+(16,16),p+(-16,16)}_{m^{ij}_k,\ast,{\gray \Delta^i_{jk},S_i},\ldots}
\ar[r]^Z
\ar@/_3ex/[rr]_z
& \left\{\calF(S)\right\}
\ar@`{p+(16,16),p+(-16,16)}_{m^{ij}_k,\ast,{\gray \Delta^i_{jk},S_i},\ldots}
\ar[r]^\bbV
& \left\{U^{\otimes S}\right\}
\ar@`{p+(16,16),p+(-16,16)}_{m^{ij}_k,\ast,{\gray \Delta^i_{jk},S_i},\ldots} \\
\text{$m^{ij}_k$: ``stitching''} &
\text{$m^{ij}_k$: ``meta-multiplication''} &
\text{$m^{ij}_k$: ``multiplication''}
} \]
The most important of these operations is the operation
$m^{ij}_k$, defined whenever $i\neq j\in S$ and $k\not\in
S\setminus\{i,j\}$. On tangles, it is ``stitching'': the operation
$\calK(S)\to\calK((S\setminus\{i,j\})\cup\{k\})$ that takes the
head of component $i$ in a tangle $K$ and stitches it to the tail
of component $j$, renaming the resulting single component $k$, as in
Figure~\ref{fig:Stitching}\footnoteT{\label{foot:careful}The careful reader will notice
that stitching is only partially defined, for the head of $i$ must lie
next to the tail of $j$ for $m^{ij}_k$ to make sense, and that it is
sometimes ill-defined, for there may be more than one path connecting
the head of $i$ with the tail of $j$. Please accept our assurances that
these issues do not lead to any difficulties, and that they are fully
resolved in Section~\ref{sec:RVT}.}. Clearly from the construction in
Aside~\ref{aside:URCMethod}, the corresponding operation on $\{U^{\otimes
S}\}$ is ``multiply tensor factor $i$ with tensor factor $j$, storing
the result in tensor factor $k$. We have a ``meta-multiplication''
operation $m^{ij}_k\colon\calF(S)\to\calF((S\setminus\{i,j\})\cup\{k\})$
which takes ``the formula for an element $\zeta$ in $U^{\otimes S}$''
to ``the formula for $m^{ij}_k(\zeta)$'', and which likewise intertwines $Z$. Namely, we
have $\bbV\act m^{ij}_k=m^{ij}\act\bbV$ and $m^{ij}_k\act Z=Z\act m^{ij}_k$.
\begin{SCfigure}
\caption{Stitching.} \label{fig:Stitching}
\input{figs/Stitching.pdf_t}
\end{SCfigure}
\item Similarly, if $S_1\cap S_2=\emptyset$, there is a ``disjoint union'' operation
$\ast\colon\calK(S_1)\times\calK(S_2)\to\calK(S_1\cup S_2)$.\footnoteT{As in
footnote~\ref{foot:careful}, there is a minor placement issue here. It is resolved in
Section~\ref{sec:RVT}.} The corresponding operation
on $\{\calU^S\}$ is the tensor product operation $\ast=\otimes\colon U^{\otimes
S_1}\times U^{\otimes S_2}\to U^{\otimes(S_1\cup S_2)}$. We ensure that there is a
compatible $\ast\colon\calF(S_1)\times\calF(S_2)\to\calF(S_1\cup S_2)$.
\item $\bbV$ is injective. A formula is determined its value.
\item The rank of $\calF(S)$ (over some ring $\calR$ of Laurent polynomials
which we will specify later) grows polynomially in the size $|S|$ of $S$, and all the
operations on $\calF(S)$ are computable using a polynomial number of ring operations.
\end{enumerate}
These five properties taken together are almost enough for what we
want. If $K$ is an $n$-crossing tangle, it be presented as some
stitching of a disjoint union of $n$ individual crossings floating
indepedently. Hence by using (1)--(3), a formula $Z(K)$ for the invariant $z(K)$ can be
computed using $O(n)$ stitchings and unions. By (4), that formula is in itself an
invariant. Finally, by (5), $Z(K)$ can be computed using a polynomial number of ring
operations (and some combinatorial overhead which amounts to much less).
To show that the computation of $Z$ is poly-time it remains to bound the complexity of
the ring elements that we encounter, and hence the complexity of ring operations among
them. This is done in {\red Section~\ref{???}}.