\section{Introduction}

In Section~\ref{ssec:Reminder} of the introduction we briefly and
schematically recall how certain algebras lead to knot invariants,
only so as to explain what exactly it is that we aim to implement
and why. Section~\ref{ssec:AbstractExpanded} of the introduction
is the abstract of this paper, expanded from one paragraph to a few
pages. Section~\ref{ssec:Plan} of the introduction is an introduction to the rest
of the paper --- a summary of what happens in it, and in what order.

\subsection{A Quick Reminder of Algebras and R-matrices} \label{ssec:Reminder}

A ``Hopf Algebra'' is a vector space $(U,+,\cdot)$ (over $\bbQ$, for
simplicity) along with a number of further operations: a ``product''
$m\colon U\otimes U\to U$, a ``coproduct'' $\Delta\colon U\to U\otimes
U$, an ``antipode'' $S\colon U\to U$, a ``unit'' $\eta\colon\bbQ\to U$ and a ``counit''
$\epsilon\colon U\to\bbQ$ (which of course are required to satisfy some axioms).  If $U$ is
also equipped with a ``braiding'' $R\in U\otimes U$ and a ``cuap element'' $C\in U$ and these
satify a few further axioms, then $U$ is a ``ribbon Hopf algebra''. It is sometimes (but not
always) useful to add to the mix a ``pairing element'' $P\in U^\ast\otimes U^{\ast}$, which is
dual to the element $R$.

Ribbon Hopf algebras are immensely useful in low dimensional topology,
as they lead to knot and tangle invariants which are well-behaved under
``strand stitching'', ``strand doubling'', ``strand reversal'', and a few
lesser operations. See e.g.~\cite[Section~4.2]{Ohtsuki:QuantumInvariants}
and our quick summary in Aside~\ref{aside:URCMethod} and in Aside~\ref{aside:TanglesAndOps}.

\begin{Aside}\fbox{\begin{minipage}{0.9\linewidth}\sl
\parpic[r]{\parbox[t]{1.9in}{\begin{center}
  \input{figs/URCMethod.pdf_t}
  \newline
  $\displaystyle z(K) = \sum_{i,j,k} b_ia_jb_kCa_ib_ja_k$
\end{center}}}
Draw $K$ as a long knot in the plane so that at each crossing the two
crossing strands are flowing up, and so that the two ends of $K$ are
flowing up.

Put a copy of $R=\sum a_i\otimes b_i$ on every positive crossing of
$K$ with the ``$a$'' side on the over-strand and the ``$b$''
side on the under-strand, labeling these $a$'s and $b$'s with distinct
indices $i,j,k,\ldots$ (similarly put copies of $R^{-1}=\sum a'_i\otimes
b'_i$ on the negative crossings; these are absent in our example). Put
a copy of $C^{\pm 1}$ on every cuap where the tangent to the knot is
pointing to the right (meaning, a $C$ on every such cup and a $C^{-1}$
on every such cap).

\picskip{2}
Form an expression $z(K)$ in $U$ by multiplying
all the $a$, $b$, $C$ letters as they are seen when traveling along
$K$ and then summing over all the indices, as shown.

If $R$ and $C$ satisfy some conditions dictated by the standard
Reidemeister moves of knot theory, the resulting $z(K)$ is a knot
invariant.

Abstractly, $z(K)$ is obtained by tensoring together several copies of
$R^{\pm 1}\in U^{\otimes 2}$ and $C^{\pm 1}\in U$ to get an intermediate
result $z_0\in U^{\otimes S}$, where $S$ is a finite set with two
elements for each crossing of $K$ and one element for each right-pointing cuap.
We then multiply the different tensor factors in $z_0$ in an order dictated by $K$
to get an output in a single copy of $U$.

\caption{The standard methodology on an example knot.} \label{aside:URCMethod}
\end{minipage}
}\end{Aside}

Yet from the perspective of topology, the algebras $U$ that
one uses seem like great wastelands with a few pearls hidden
within. From the perspective of Aside~\ref{aside:URCMethod} and
Aside~\ref{aside:TanglesAndOps} the vector space structure of $U$
is completely irrelevant as the operations of addition ($+$) and
multiplication by a scalar ($\cdot$) are never used. All that matters are
those elements (the ``pearls'') within tensor powers $U^{\otimes S}$ of $U$
that can be written using the ``generators'' $R$ and $C$, using tensor
products $U^{\otimes S_1}\times U^{\otimes S_2}\to U^{\otimes(S_1\sqcup
S_2)}$, and using the multiplication $m$ (extended to tensor powers) {\em yet without using $+$
and $\cdot$}\footnoteT{It does not matter whther or not $\Delta$, $S$, $\eta$, and $\epsilon$
are used for the generation of the ``pearls'', as the axioms of a ribbon Hopf algebra imply
that anything that can be generated with them can also be generated without them.}.

{\red MORE.}

\subsection{An Expansion of the Abstract} \label{ssec:AbstractExpanded}

{\red MORE.}

\subsection{Plan of the paper.} \label{ssec:Plan}

{\red MORE.}

\subsection{Acknowledgement}

{\red MORE.}