===== 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''.}