===== recycled on Tue Jun 27 09:32:31 EDT 2017 by drorbn on Ubuntu-1404 ======


\begin{figure}[h]
\begin{center}
\parbox[t]{1.9in}{\begin{center}
\raisebox{7mm}{\imagetop{\input{figs/URCMethod.pdf_t}}}
\newline
$\displaystyle z(K) = \sum_{i,j,k} b_ia_jb_kCa_ib_ja_k$
\end{center}}
\parbox[t]{4.5in}{\small\sl
Draw $K$ in the plane so that at each crossing the two crossing strands
are pointed 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).

If $K$ is a (long) knot, 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.

When $K$ is a tangle with $S$ strands, carry out the multiplications along
each strand separately in a different tensor-copy of $U$, to get $z(K)\in
U^{\otimes S}$.

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

}
\end{center}
\caption{The standard methodology on an example knot.} \label{fig:URCMethod}
\end{figure}

===== recycled on Tue Jun 27 10:25:10 EDT 2017 by drorbn on Ubuntu-1404 ======


\begin{figure}[h]\begin{minipage}{\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 $K$ in the plane so that at each crossing the two crossing strands
are pointed 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).

If $K$ is a (long) knot, 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.

\picskip{1}
When $K$ is a tangle with $S$ strands, carry out the multiplications along
each strand separately in a different tensor-copy of $U$, to get $z(K)\in
U^{\otimes S}$.

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

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

===== recycled on Tue Jun 27 10:37:13 EDT 2017 by drorbn on Ubuntu-1404 ======

\footnoteT{A ``tangle'' for current purposes is a
multi-component knot whose components (``strands'') are all (oriented)
intervals (i.e., not circles) and are in a bijection with some finite
set $S$ of ``strand labels''.}