===== recycled on Tue 31 Dec 2024 06:54:09 AM EST by drorbn on Ubuntu-on-X2 ======


\includegraphics[width=0.19\linewidth]{figs/PP300.png}
\includegraphics[width=0.19\linewidth]{figs/PP301.png}
\includegraphics[width=0.19\linewidth]{figs/PP302.png}
\includegraphics[width=0.19\linewidth]{figs/PP303.png}
\includegraphics[width=0.19\linewidth]{figs/PP304.png}
\includegraphics[width=0.19\linewidth]{figs/PP305.png}
\includegraphics[width=0.19\linewidth]{figs/PP306.png}
\includegraphics[width=0.19\linewidth]{figs/PP307.png}
\includegraphics[width=0.19\linewidth]{figs/PP308.png}
\includegraphics[width=0.19\linewidth]{figs/PP309.png}
\includegraphics[width=0.19\linewidth]{figs/PP310.png}
\includegraphics[width=0.19\linewidth]{figs/PP311.png}
\includegraphics[width=0.19\linewidth]{figs/PP312.png}
\includegraphics[width=0.19\linewidth]{figs/PP313.png}
\includegraphics[width=0.19\linewidth]{figs/PP317.png}

===== recycled on Tue 31 Dec 2024 07:31:33 AM EST by drorbn on Ubuntu-on-X2 ======


The formulas for $\theta$ depend on three fixed polynomials $F_1(c)$,
$F_2(c_0,c_1)$ and $F_3(\varphi,k)$ in the $g_{\nu\alpha\beta}$'s,
which we admit, are rather ugly. So we prefer to assert their existance
and postpone displaying them to a few paragraphs later.

===== recycled on Thu 02 Jan 2025 04:08:35 AM EST by drorbn on Ubuntu-on-X2 ======


\begin{multline} \label{eq:F1}
  F_1(c) = s
    \left[ 1/2 - g_{3ii} +  T_2^s g_{1ii} g_{2ji} - T_2^s g_{3jj} g_{2ji}
      - (T_2^s-1) g_{3ii} g_{2ji} \right. \\
    \left. + (T_3^s-1) g_{2ji} g_{3ji} - g_{1ii} g_{2jj} + 2 g_{3ii} g_{2jj}
      + g_{1ii} g_{3jj} - g_{2ii} g_{3jj} \right] \\
  + \frac{s}{T_2^s-1}
    \left[
      (T_1^s-1)T_2^s \left( g_{3jj} g_{1ji} - g_{2jj} g_{1ji} + T_2^s g_{1ji} g_{2ji} \right) \right. \\
      + (T_3^s-1) \left( g_{3ji} - T_2^s g_{1ii} g_{3ji} + g_{2ij} g_{3ji}
      + (T_2^s-2) g_{2jj} g_{3ji} \right) \\
    \left. - (T_1^s-1) (T_2^s+1) (T_3^s-1) g_{1ji} g_{3ji} \right]
\end{multline}
\begin{equation} \label{eq:F2}
  F_2(c_0,c_1) =
    \frac{s_1 (T_1^{s_0}-1) (T_3^{s_1}-1) g_{1j_1i_0} g_{3j_0i_1}}{T_2^{s_1}-1}
    \left(T_2^{s_0} g_{2i_1i_0}+g_{2j_1j_0} - T_2^{s_0} g_{2j_1i_0}-g_{2i_1j_0} \right)
\end{equation}
\begin{equation} \label{eq:F3} F_3(\varphi,k) = \varphi(g_{3kk}-1/2) \end{equation}

===== recycled on Sat 11 Jan 2025 02:07:36 PM EST by drorbn on Ubuntu-on-X2 ======


\begin{technicality} \label{tech:nonseq}
Some Reidemeister moves create or lose an edge 
and to avoid the need for renumbering it is beneficial to also allow
labelling the edges with non-consecutive labels. Hence we allow that, and
write $\ip$ for the successor of the label $i$ along the knot, and $\ipp$
for the successor of $\ip$ (these are $i+1$ and $i+2$ if the labelling is
by consecutive integers). Also, by convention ``$1$'' will always refer to the label of
the first edge, and ``$2n+1$'' will always refer to the label of the last. \endpar{\ref{tech:nonseq}}
\end{technicality}

===== recycled on Sat 11 Jan 2025 02:07:55 PM EST by drorbn on Ubuntu-on-X2 ======


\begin{clarification} In Theorem~\ref{thm:RelativeInvariant}, $\alpha$
and $\beta$ stand for {\em edges} and not for their serial numbers, which
may change if edges are numbered sequencialy when a Reidemeister move
that changes the overall number of crossings is performed. It is for this reason that we've introdcued
Technicality~\ref{tech:nonseq}.
\end{clarification}

===== recycled on Sat 11 Jan 2025 02:07:59 PM EST by drorbn on Ubuntu-on-X2 ======


\begin{proviso} A further minor issue arrises if $\alpha$ and $\beta$
are distinct before a Reidemeister move is performed but become the
same after the move (see an illustration on the right).
\end{proviso}