\noindent{\em Proof.} For R2c$^+$ we follow the same logic as in the proof of Proposition~\ref{prop:R3}, as simplified by Remark~\ref{rem:E}. We start with the figure that replaces Figure~\ref{fig:R3} (note the null vertices in $D^r$ and their minimal effect as in Lemma~\ref{lem:NullVertices} and Remark~\ref{rem:F4Null}):
\[ {
  \def\i{{$i$}} \def\j{{$j$}} \def\k{{$k$}} \def\m{{$m$}} \def\n{{$n$}} \def\s{{$s$}}
  \input{figs/R2cE.pdf_t} 
} \]

As in Remark~\ref{rem:E}, we let $E^l$ and $E^r$ be the sums corresponding to the diagrams $D^l$ and $D^r$ above:
\[
  E^l = \sum_{\makebox[0pt]{$\scriptstyle c\in\{c^l_1,c^l_2,c_y\}$}} F^l_1(c)
    \ +\ \sum_{\makebox[0pt]{$\scriptstyle c_0,c_1\in\{c^l_1,c^l_2,c_y\}$}} F^l_2(c_0,c_1)
    \ +\ F^l_3(\jp)|_{\varphi_{\jp}=1},
  \quad
    E^r = F^r_1(c_y) \ +\ F^r_2(c_y,c_y) \ +\ F^r_3(\jp)|_{\varphi_{\jp}=1}.
\]
We need to show that $E^l=E^r$ after all relevant $g$-rules are applied to both sides.

To compute these $E$ sums we first have to extend the \verb$ESum$ routine to accept also a list $R$ of pairs $(\varphi,k)$ of the form (rotation number, edge label):

\nbpdfInput{Snips/Implementation/44.pdf}
We then compute $E^l$ (and apply the relevant $g$-rules) by calling \verb$ESum$ with crossings $(-1,i,\jp)$, $(1,\ip,j)$, and $(s,m,n)$, and a rotation number of $1$ on edge $\jp$:

\nbpdfInput{Snips/Implementation/45.pdf}
\nbpdfOutput{Snips/Implementation/46.pdf}
The computation of $E^r$ is simpler, as it only involves the generic $(s,m,n)$ and the rotation $(1,\jp)$. We implement the $g$-rules for null vertices as in Equations~\eqref{eq:NullCarRules} and~\eqref{eq:NullCounterRules}, compute $E^r$, and then compare $E^l$ with $E^r$ to conclude the invariance under R2c$^+$:

\nbpdfInput{Snips/Implementation/47.pdf}
\nbpdfInput{Snips/Implementation/48.pdf}
\nbpdfOutput{Snips/Implementation/49.pdf}
For R2c$^-$ we allow ourselves to be even more concise:

\parpic[r]{ \def\i{{$i$}} \def\j{{$j$}}  \input{figs/uR2cm.pdf_t} }
\picskip{1}

\nbpdfInput{Snips/Implementation/50.pdf}
\nbpdfOutput{Snips/Implementation/51.pdf}
\endpar{\ref{prop:R2c}}