[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 1, 2 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (2.000000, 0.000000) arc (0.000000:180.000000:1.250000);
[] (-0.500000, 0.000000) arc (180.000000:360.000000:0.500000);
[] (0.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (1.500000, 0.000000) arc (0.000000:-180.000000:1.100000);
[] (-0.700000, 0.000000) arc (180.000000:0.000000:1.600000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:2.000000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (5.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:180.000000:0.900000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:0.400000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:0.400000);
[] (4.300000, 0.000000) arc (180.000000:360.000000:0.850000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 5, 6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (2.000000, 0.000000) arc (0.000000:180.000000:1.250000);
[] (-0.500000, 0.000000) arc (180.000000:360.000000:0.500000);
[] (0.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (1.500000, 0.000000) arc (0.000000:-180.000000:1.100000);
[] (-0.700000, 0.000000) arc (180.000000:0.000000:1.600000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:2.000000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.000000);
[] (4.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (3.500000, 0.000000) arc (0.000000:180.000000:0.400000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:1.000000);
[] (4.700000, 0.000000) arc (180.000000:0.000000:0.650000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (2.000000, 0.000000) arc (0.000000:180.000000:1.250000);
[] (-0.500000, 0.000000) arc (180.000000:360.000000:0.500000);
[] (0.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (1.500000, 0.000000) arc (0.000000:-180.000000:1.100000);
[] (-0.700000, 0.000000) arc (180.000000:0.000000:1.600000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:2.000000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.900000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:0.400000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) – (1.000000, 0.000000) ;
[] (1.000000, 0.000000) arc (0.000000:180.000000:0.750000);
[] (-0.500000, 0.000000) arc (180.000000:360.000000:1.000000);
[] (1.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (2.500000, 0.000000) arc (0.000000:-180.000000:1.600000);
[] (-0.700000, 0.000000) arc (180.000000:0.000000:1.700000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:1.900000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.850000);
[] (2.800000, 0.000000) arc (180.000000:360.000000:0.350000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) – (1.000000, 0.000000) ;
[] (1.000000, 0.000000) arc (0.000000:180.000000:0.750000);
[] (-0.500000, 0.000000) arc (180.000000:360.000000:1.500000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (3.500000, 0.000000) arc (0.000000:-180.000000:2.100000);
[] (-0.700000, 0.000000) arc (180.000000:0.000000:1.100000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:0.400000);
[] (2.300000, 0.000000) arc (180.000000:0.000000:0.700000);
[] (3.700000, 0.000000) arc (180.000000:360.000000:1.400000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.350000);
[] (3.800000, 0.000000) arc (180.000000:360.000000:0.350000);
[] (4.500000, 0.000000) arc (180.000000:0.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) – (1.000000, 0.000000) ;
[] (1.000000, 0.000000) arc (180.000000:360.000000:0.750000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (3.500000, 0.000000) arc (0.000000:-180.000000:1.500000);
[] (0.500000, 0.000000) arc (180.000000:0.000000:1.600000);
[] (3.700000, 0.000000) arc (180.000000:360.000000:1.400000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.350000);
[] (3.800000, 0.000000) arc (180.000000:360.000000:0.350000);
[] (4.500000, 0.000000) arc (180.000000:0.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (3.000000, 0.000000) arc (0.000000:180.000000:1.000000);
[] (1.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (2.500000, 0.000000) arc (0.000000:-180.000000:1.000000);
[] (0.500000, 0.000000) arc (180.000000:0.000000:1.500000);
[] (3.500000, 0.000000) arc (180.000000:360.000000:1.500000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.000000);
[] (4.500000, 0.000000) arc (180.000000:360.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_3 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\left( \Gamma_{C_2}^{-1}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{L_1}\Gamma_{C_2'}\Gamma_{L_1}\Gamma_{C_2'}^{-1}\Gamma_{L_1}^{-1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{C_2}  \right) \left( \Gamma_{L_3}  \right) \left( \Gamma_{C_2}^{-1}\Gamma_{C_1}\Gamma_{C_2}  \right)& = \\
\left( \Gamma_{L_3}  \right) \left( \Gamma_{C_2}^{-1}\Gamma_{C_1}\Gamma_{C_2}  \right) \left( \Gamma_{C_2}^{-1}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{L_1}\Gamma_{C_2'}\Gamma_{L_1}\Gamma_{C_2'}^{-1}\Gamma_{L_1}^{-1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{C_2}  \right)& = \\
\left( \Gamma_{C_2}^{-1}\Gamma_{C_1}\Gamma_{C_2}  \right) \left( \Gamma_{C_2}^{-1}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{L_1}\Gamma_{C_2'}\Gamma_{L_1}\Gamma_{C_2'}^{-1}\Gamma_{L_1}^{-1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{C_2}  \right) \left( \Gamma_{L_3}  \right)
\end{split}
\end{equation*}

§.§.§ Vertex Number 15

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) – (5.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:1.250000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:0.500000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}\mathbb{R}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:1.250000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:0.500000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:1.250000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:0.500000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:1.250000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:1.000000);
[] (4.500000, 0.000000) arc (0.000000:180.000000:0.900000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:0.400000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:0.400000);
[] (4.300000, 0.000000) arc (0.000000:-180.000000:0.850000);
[] (2.600000, 0.000000) arc (180.000000:0.000000:1.200000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 5, 6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:1.250000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:1.500000);
[] (5.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (3.500000, 0.000000) arc (0.000000:180.000000:0.400000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:1.000000);
[] (4.700000, 0.000000) arc (180.000000:0.000000:0.300000);
[] (5.300000, 0.000000) arc (0.000000:-180.000000:1.350000);
[] (2.600000, 0.000000) arc (180.000000:0.000000:0.700000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 6, 7 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:1.250000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:2.000000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (5.500000, 0.000000) arc (0.000000:-180.000000:1.000000);
[] (3.500000, 0.000000) arc (0.000000:180.000000:0.400000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:1.500000);
[] (5.700000, 0.000000) arc (180.000000:0.000000:0.300000);
[] (6.300000, 0.000000) arc (0.000000:-180.000000:1.850000);
[] (2.600000, 0.000000) arc (180.000000:0.000000:0.700000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:1.250000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:2.000000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (5.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:180.000000:0.900000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:0.400000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:0.400000);
[] (4.300000, 0.000000) arc (180.000000:360.000000:0.700000);
[] (5.700000, 0.000000) arc (180.000000:0.000000:0.300000);
[] (6.300000, 0.000000) arc (0.000000:-180.000000:1.150000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 1, 2 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (0.000000:-180.000000:1.000000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:1.500000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:2.000000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (5.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:180.000000:0.900000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:0.400000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:0.400000);
[] (4.300000, 0.000000) arc (180.000000:360.000000:0.700000);
[] (5.700000, 0.000000) arc (180.000000:0.000000:0.300000);
[] (6.300000, 0.000000) arc (0.000000:-180.000000:1.150000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 5, 6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (0.000000:-180.000000:1.000000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:1.500000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:2.000000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.000000);
[] (4.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (3.500000, 0.000000) arc (0.000000:180.000000:0.400000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:1.000000);
[] (4.700000, 0.000000) arc (180.000000:0.000000:0.800000);
[] (6.300000, 0.000000) arc (0.000000:-180.000000:0.650000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (0.000000:-180.000000:1.000000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:1.500000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:2.000000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.900000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:0.400000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:1.400000);
[] (6.300000, 0.000000) arc (0.000000:-180.000000:0.650000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:360.000000:0.750000);
[] (1.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (2.500000, 0.000000) arc (0.000000:-180.000000:1.500000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:1.600000);
[] (2.700000, 0.000000) arc (180.000000:360.000000:1.900000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.850000);
[] (2.800000, 0.000000) arc (180.000000:360.000000:0.350000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:1.400000);
[] (6.300000, 0.000000) arc (0.000000:-180.000000:0.650000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:360.000000:1.250000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (3.500000, 0.000000) arc (0.000000:-180.000000:2.000000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:0.400000);
[] (2.300000, 0.000000) arc (180.000000:0.000000:0.700000);
[] (3.700000, 0.000000) arc (180.000000:360.000000:1.400000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.350000);
[] (3.800000, 0.000000) arc (180.000000:360.000000:0.350000);
[] (4.500000, 0.000000) arc (180.000000:0.000000:0.900000);
[] (6.300000, 0.000000) arc (0.000000:-180.000000:0.650000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (3.500000, 0.000000) arc (0.000000:-180.000000:1.500000);
[] (0.500000, 0.000000) arc (180.000000:0.000000:1.600000);
[] (3.700000, 0.000000) arc (180.000000:360.000000:1.400000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.350000);
[] (3.800000, 0.000000) arc (180.000000:360.000000:0.350000);
[] (4.500000, 0.000000) arc (180.000000:0.000000:0.900000);
[] (6.300000, 0.000000) arc (0.000000:-180.000000:0.650000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (3.000000, 0.000000) arc (0.000000:-180.000000:1.250000);
[] (0.500000, 0.000000) arc (180.000000:0.000000:1.500000);
[] (3.500000, 0.000000) arc (180.000000:360.000000:1.500000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.000000);
[] (4.500000, 0.000000) arc (180.000000:360.000000:0.500000);
[] (5.500000, 0.000000) arc (180.000000:0.000000:0.400000);
[] (6.300000, 0.000000) arc (0.000000:-180.000000:1.150000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_3 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\Gamma_{L_1}\Gamma_{C_2'}^{-1}\Gamma_{L_1}^{-1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{C_1}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{L_1}\Gamma_{C_2'}\Gamma_{L_1}^{-1}  = \Gamma_{C_1'}
\end{split}
\end{equation*}

§.§.§ Raw relations and simplifications

\begin{equation}\label{arrangement1_auto_calc_vert_1_rel_1}
\begin{split}
\Gamma_{C_1}  = \Gamma_{C_1'} ,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_2_rel_1}
\begin{split}
\left[\Gamma_{C_2}^{-1}\Gamma_{C_1}\Gamma_{C_2} , \Gamma_{L_3}\Gamma_{L_2}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_2_rel_2}
\begin{split}
\left[\Gamma_{L_3} , \Gamma_{L_2}\Gamma_{C_2}^{-1}\Gamma_{C_1}\Gamma_{C_2}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_4_rel_1}
\begin{split}
\Gamma_{C_1'}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{L_2}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{L_2}^{-1}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{C_1'}^{-1}  = \Gamma_{C_2'} ,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_5_rel_1}
\begin{split}
\left[\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1} , \Gamma_{C_2}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_6_rel_1}
\begin{split}
\left\{\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{C_1'}^{-1}\Gamma_{C_2'}\Gamma_{C_1'}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3} , \Gamma_{L_2}  \right\}=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_7_rel_1}
\begin{split}
\left[\Gamma_{C_2'}\Gamma_{C_1'}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{L_2}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{C_1'}^{-1}\Gamma_{C_2'}^{-1} , \Gamma_{C_1'}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_8_rel_1}
\begin{split}
\left\{\Gamma_{C_1} , \Gamma_{C_2}  \right\}=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_9_rel_1}
\begin{split}
\left\{\Gamma_{C_2'} , \Gamma_{C_1'}  \right\}=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_10_rel_1}
\begin{split}
\left[\Gamma_{C_2'}\Gamma_{C_1'}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{L_2}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{C_1'}^{-1}\Gamma_{C_2'}^{-1} , \Gamma_{L_1}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_11_rel_1}
\begin{split}
\left[\Gamma_{C_2'}\Gamma_{C_1'}\Gamma_{C_2'}^{-1} , \Gamma_{L_1}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_12_rel_1}
\begin{split}
\left\{\Gamma_{C_2'} , \Gamma_{L_1}  \right\}=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_13_rel_1}
\begin{split}
\left[\Gamma_{C_2} , \Gamma_{L_3}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_14_rel_1}
\begin{split}
\Gamma_{L_1}^{-1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{C_2}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{L_1}  = \Gamma_{C_2'} ,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_15_rel_1}
\begin{split}
\left[\Gamma_{C_2}^{-1}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{L_1}\Gamma_{C_2'}\Gamma_{L_1}\Gamma_{C_2'}^{-1}\Gamma_{L_1}^{-1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{C_2} , \Gamma_{L_3}\Gamma_{C_2}^{-1}\Gamma_{C_1}\Gamma_{C_2}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_15_rel_2}
\begin{split}
\left[\Gamma_{L_3} , \Gamma_{C_2}^{-1}\Gamma_{C_1}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{L_1}\Gamma_{C_2'}\Gamma_{L_1}\Gamma_{C_2'}^{-1}\Gamma_{L_1}^{-1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{C_2}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_vert_16_rel_1}
\begin{split}
\Gamma_{L_1}\Gamma_{C_2'}^{-1}\Gamma_{L_1}^{-1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{C_1}\Gamma_{L_3}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{L_1}\Gamma_{C_2'}\Gamma_{L_1}^{-1}  = \Gamma_{C_1'} ,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_auto_calc_projective_rel}
\begin{split}
\Gamma_{L_1}\Gamma_{C_2'}\Gamma_{C_1'}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{L_2} =e.
\end{split}
\end{equation}

Here are the computations of group $ G_5 := \pcpt{\mathcal{B}_5} / \langle \Gamma_X^2 \; | \; X\subseteq \mathcal{B}_5 \rangle $:

The quotient $ G_5 := \pcpt{\mathcal{B}_5} / \langle \Gamma_X^2 \; | \; X\subseteq \mathcal{B}_5 \rangle $ is generated by $ \Gamma_{C_1}, \Gamma_{C_2}, \Gamma_{L_1}, \Gamma_{L_2} $, and $ \Gamma_{L_3} $, subject to the following relations:

\begin{equation}\label{arrangement1_final_squares}
\Gamma_{C_1}^2 = \Gamma_{C_2}^2 = \Gamma_{L_1}^2 = \Gamma_{L_2}^2 = \Gamma_{L_3}^2 = e,
\end{equation}

\begin{equation}\label{arrangement1_final1}
[\Gamma_{L_3},\Gamma_{L_1}] = [\Gamma_{L_3}, \Gamma_{L_2}] = [\Gamma_{L_3}, \Gamma_{C_1}] = [\Gamma_{L_3}, \Gamma_{C_2}] = [\Gamma_{L_1}, \Gamma_{L_2}] = [\Gamma_{C_1}, \Gamma_{L_1}] = [\Gamma_{C_1}, \Gamma_{L_2}] = e,
\end{equation}

\begin{equation}\label{arrangement1_final2}
[\Gamma_{L_1}, \Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}] = [\Gamma_{L_2}, \Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}] = [\Gamma_{C_2}, \Gamma_{L_1}\Gamma_{L_2}] = e,
\end{equation}

\begin{equation}\label{arrangement1_final3}
\{\Gamma_{C_1}, \Gamma_{C_2}\} = \{ \Gamma_{C_2}, \Gamma_{L_1} \} = \{ \Gamma_{C_2}, \Gamma_{L_2}\} = e,
\end{equation}

\begin{equation}\label{arrangement1_final4}
\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{L_2} = e.
\end{equation}

In particular, $ \Gamma_{L_3} $ is in the center of $ G_5 $.

After the substitutions $ \Gamma_{C_1'} = \Gamma_{C_1} $ and $ \Gamma_{C_2'} = \Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1} $ we derive that $ G_5 $ is generated by $ \Gamma_{C_1},\Gamma_{C_2},\Gamma_{L_1},\Gamma_{L_2} $, and $ \Gamma_{L_3} $, subject to:

\begin{equation}\label{arrangement1_simpl_1}
\begin{split}
\left[\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2} , \Gamma_{L_3}\Gamma_{L_2}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_2}
\begin{split}
\left[\Gamma_{L_3} , \Gamma_{L_2}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_3}
\begin{split} \Gamma_{C_2}\Gamma_{L_3}\Gamma_{L_2}\Gamma_{L_3}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{L_2}\Gamma_{L_3}\Gamma_{C_2}  = \Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1} ,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_4}
\begin{split}
\left[\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{C_2}\Gamma_{C_1} , \Gamma_{C_2}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_5}
\begin{split} \left\{\Gamma_{L_3}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_3} , \Gamma_{L_2}  \right\}=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_6}
\begin{split} \left[\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{L_2}\Gamma_{L_3}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1} , \Gamma_{C_1}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_7}
\begin{split}
\left\{\Gamma_{C_1} , \Gamma_{C_2}  \right\}=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_8}
\begin{split}
\left\{\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1} , \Gamma_{C_1}  \right\}=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_9}
\begin{split} \left[\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{L_2}\Gamma_{L_3}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1} , \Gamma_{L_1}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_10}
\begin{split} \left[\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1} , \Gamma_{L_1}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_11}
\begin{split}
\left\{\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1} , \Gamma_{L_1}  \right\}=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_12}
\begin{split}
\left[\Gamma_{C_2} , \Gamma_{L_3}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_13}
\begin{split}
\left[\Gamma_{C_2}\Gamma_{L_3}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{L_3}\Gamma_{C_2} , \Gamma_{L_3}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_14}
\begin{split}
\left[\Gamma_{L_3} , \Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_3}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{L_3}\Gamma_{C_2}  \right]=e,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_15}
\begin{split}
\Gamma_{C_1}\Gamma_{L_3}\Gamma_{C_1}\Gamma_{L_3}\Gamma_{C_1}  = \Gamma_{C_1} ,
\end{split}
\end{equation}

\begin{equation}\label{arrangement1_simpl_16}
\begin{split}
\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_3}\Gamma_{L_2} =e.
\end{split}
\end{equation}

From (<ref>) we get that $ \Gamma_{L_3} $ commutes with $ \Gamma_{C_1} $.
Substituting this, together with (<ref>), into (<ref>) and (<ref>) gives us that $ \Gamma_{L_3} $ commutes with $ \Gamma_{L_2} $ and $ \Gamma_{L_1} $, so $ \Gamma_{L_3} $ is in the center of $ G_5 $.
Omitting $ \Gamma_{L_3} $ from (<ref>) yields $ [\Gamma_{C_1}, \Gamma_{L_1}] = e $.
By applying this we get $ \{ \Gamma_{L_1}, \Gamma_{C_2} \} = e $ from (<ref>) and $ [\Gamma_{L_1}, \Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}]=e $ from (<ref>).
By isolating $ \Gamma_{L_2} $ in (<ref>) we get

\begin{align*}
e & = [ \Gamma_{L_2}, \Gamma_{L_3}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_3} ] = \\
& = [ \Gamma_{L_2}, \Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2} ] = \\
& = [ \Gamma_{L_2}, \Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2} ] = \\
& = [ \Gamma_{L_2}, \Gamma_{L_1} ],
\end{align*}

where in the first equality we used the fact that $ \Gamma_{L_3} $ is in the center, as well as the fact that $ \Gamma_{C_1} $ and $ \Gamma_{L_1} $ commute; in the second equality we used $ \{ \Gamma_{C_2}, \Gamma_{L_1} \} = e $; and in the third equality we used $ [\Gamma_{L_1}, \Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}] = e $.

By isolating $ \Gamma_{L_2} $ in (<ref>) we get

\begin{align*}
e &= [\Gamma_{L_2}, \Gamma_{L_3}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_3}] = \\
& =  [\Gamma_{L_2}, \Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_2}] = \\
& =  [\Gamma_{L_2}, \Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_2}] = \\
& =  [\Gamma_{L_2}, \Gamma_{C_1}],
\end{align*}

where again we used known commutation relations in the first equality, $ {\{\Gamma_{C_1}, \Gamma_{C_2}\} = e} $ in the second equality, and $ [\Gamma_{L_1}, \Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}] = e $ in the third equality.

Next we deal with (<ref>). By eliminating $ \Gamma_{L_3} $ and rearranging, we get

\begin{align*}
& \Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2} = \\
=& \Gamma_{L_2}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_2} = \\
=& \Gamma_{L_2}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_2} = \\
=& \Gamma_{L_2}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_2} = \\
=& \Gamma_{L_2}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{L_2} = \\
=& \Gamma_{L_2}\Gamma_{L_1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{L_1}\Gamma_{L_2}.
\end{align*}

By $ \{ \Gamma_{C_1}, \Gamma_{C_2} \} =e $, the left side of the above equation is equal to $ \Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1} $, and by using the commutation relations $ [\Gamma_{C_1},\Gamma_{L_1}]=[\Gamma_{C_1}, \Gamma_{L_2}]=e $, the above equation is equivalent to $ [\Gamma_{C_2}, \Gamma_{L_1}\Gamma_{L_2}]=e $.

From (<ref>) we get $ [\Gamma_{L_2}, \Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}]=e $. Using this together with $ \{ \Gamma_{C_2}, \Gamma_{L_1} \}=e $ and $ [\Gamma_{L_1}, \Gamma_{L_2}]=e $, we can rearrange (<ref>) and derive

\begin{align*}
e & = \{ \Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_2}\Gamma_{C_2}\Gamma_{C_1}, \Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_1} \} = \\
& = \{ \Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_2}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}, \Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_1}\Gamma_{C_2} \} = \\
& = \{ \Gamma_{L_2}, \Gamma_{L_1}\Gamma_{C_2}\Gamma_{L_1} \} =
\{ \Gamma_{L_2}, \Gamma_{C_2} \}.
\end{align*}

Combining those derivations we get the desired result.

§.§ $ \pcpt{\mathcal{B}_6} $

[ , scale=1]
[rotate around=90.000000:(0.000000,0.000000)] (0.000000,0.000000) ellipse (4.000000 and 4.000000);
(C_1_naming_node) at (3.000000,3.000000) $ C_1 $;
[] (4.800000, -3.634924) – (-0.649056, 9.198599) node[near end, above] $L_2$;
[rotate around=-89.000000:(-0.000000,-0.000000)] (-0.000000,-0.000000) ellipse (4.000000 and 2.800000);
(C_2_naming_node) at (-3.069810,-2.999391) $ C_2 $;
[] (4.800000, -2.124063) – (-4.800000, 6.063912) node[very near end, below] $L_4$;
[] (0.321088, 9.198599) – (-4.800000, -4.138215) node[very near end, below] $L_1$;
[fill=gray] (4.000000, -0.000000) circle (5.000000pt);
[label=above:1] (sing_pnt_1) at (4.000000,-0.000000) ;
[fill=gray] (3.794280, -1.266270) circle (5.000000pt);
[label=above:2] (sing_pnt_2) at (3.794280,-1.266270) ;
[fill=gray] (2.800444, -0.050845) circle (5.000000pt);
[label=above:3] (sing_pnt_3) at (2.800444,-0.050845) ;
[fill=gray] (2.789179, -0.409005) circle (5.000000pt);
[label=above:4] (sing_pnt_4) at (2.789179,-0.409005) ;
[fill=gray] (2.389597, 2.042015) circle (5.000000pt);
[label=above:5] (sing_pnt_5) at (2.389597,2.042015) ;
[fill=gray] (1.724112, 3.609354) circle (5.000000pt);
[label=above:6] (sing_pnt_6) at (1.724112,3.609354) ;
[fill=gray] (0.069810, -3.999391) circle (5.000000pt);
[label=above:7] (sing_pnt_7) at (0.069810,-3.999391) ;
[fill=gray] (-0.069810, 3.999391) circle (5.000000pt);
[label=above:8] (sing_pnt_8) at (-0.069810,3.999391) ;
[fill=gray] (-0.139619, 7.998782) circle (5.000000pt);
[label=above:9] (sing_pnt_9) at (-0.139619,7.998782) ;
[fill=gray] (-1.587276, 3.323734) circle (5.000000pt);
[label=above:10] (sing_pnt_10) at (-1.587276,3.323734) ;
[fill=gray] (-1.849026, 3.546985) circle (5.000000pt);
[label=above:11] (sing_pnt_11) at (-1.849026,3.546985) ;
[fill=gray] (-2.459407, 1.957376) circle (5.000000pt);
[label=above:12] (sing_pnt_12) at (-2.459407,1.957376) ;
[fill=gray] (-2.800444, 0.050845) circle (5.000000pt);
[label=above:13] (sing_pnt_13) at (-2.800444,0.050845) ;
[fill=gray] (-3.747776, -1.397917) circle (5.000000pt);
[label=above:14] (sing_pnt_14) at (-3.747776,-1.397917) ;
[fill=gray] (-4.000000, 0.000000) circle (5.000000pt);
[label=above:15] (sing_pnt_15) at (-4.000000,0.000000) ;

Monodromy table

Vertex number    Vertex description    Skeleton    Diffeomorphism
1    $C_1$ branch    $ \langle 3 - 4 \rangle $    $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $
2    Node between $C_1$, $L_4$ and $L_2$    $ \langle 1 - 2 - 3 \rangle $    $ \Delta \langle 1,2,3 \rangle $
3    $C_2$ branch    $ \langle 3 - 4 \rangle $    $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $
4    Node between $C_2$ and $L_4$    $ \langle 2 - 3 \rangle $    $ \Delta\langle 2, 3 \rangle $
5    Tangency between $L_2$ and $C_2$    $ \langle 4 - 5 \rangle $    $ \Delta^{2}\langle 4, 5 \rangle $
6    Node between $C_1$ and $L_2$    $ \langle 5 - 6 \rangle $    $ \Delta\langle 5, 6 \rangle $
7    Tangency between $C_1$ and $C_2$    $ \langle 1 - 2 \rangle $    $ \Delta^{2}\langle 1, 2 \rangle $
8    Tangency between $C_1$ and $C_2$    $ \langle 4 - 5 \rangle $    $ \Delta^{2}\langle 4, 5 \rangle $
9    Node between $L_1$ and $L_2$    $ \langle 6 - 7 \rangle $    $ \Delta\langle 6, 7 \rangle $
10    Node between $C_2$ and $L_4$    $ \langle 3 - 4 \rangle $    $ \Delta\langle 3, 4 \rangle $
11    Node between $L_1$, $C_1$ and $L_4$    $ \langle 4 - 5 - 6 \rangle $    $ \Delta \langle 4,5,6 \rangle $
12    Tangency between $L_1$ and $C_2$    $ \langle 3 - 4 \rangle $    $ \Delta^{2}\langle 3, 4 \rangle $
13    $C_2$ branch    $ \langle 2 - 3 \rangle $    $ \Delta_{I_{2}\mathbb{R}}^{1/2}\langle 2 \rangle $
14    Node between $L_1$ and $C_1$    $ \langle 1 - 2 \rangle $    $ \Delta\langle 1, 2 \rangle $
15    $C_1$ branch    $ \langle 2 - 3 \rangle $    $ \Delta_{I_{4}I_{2}}^{1/2}\langle 2 \rangle $

§.§.§ Vertex Number 1

[] (3.000000, 0.000000) – (4.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\Gamma_{C_1}  = \Gamma_{C_1'}
\end{split}
\end{equation*}

§.§.§ Vertex Number 2

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) – (1.000000, 0.000000) ;
[] (1.000000, 0.000000) – (2.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (0.000000, 0.000000) – (1.000000, 0.000000) ;
[] (1.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\left( \Gamma_{C_2}^{-1}\Gamma_{C_1}\Gamma_{C_2}  \right) \left( \Gamma_{L_4}  \right) \left( \Gamma_{L_2}  \right)& = \\
\left( \Gamma_{L_4}  \right) \left( \Gamma_{L_2}  \right) \left( \Gamma_{C_2}^{-1}\Gamma_{C_1}\Gamma_{C_2}  \right)& = \\
\left( \Gamma_{L_2}  \right) \left( \Gamma_{C_2}^{-1}\Gamma_{C_1}\Gamma_{C_2}  \right) \left( \Gamma_{L_4}  \right)
\end{split}
\end{equation*}

§.§.§ Vertex Number 3

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (1.500000, 0.000000) arc (180.000000:0.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (0.000000:180.000000:1.250000);
[] (0.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:2.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (2.000000, 0.000000) arc (180.000000:360.000000:0.750000);
[] (3.500000, 0.000000) arc (0.000000:180.000000:1.500000);
[] (0.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:2.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\Gamma_{C_1'}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_4}\Gamma_{L_2}\Gamma_{L_4}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{C_2}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_4}\Gamma_{L_2}^{-1}\Gamma_{L_4}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{C_1'}^{-1}  = \Gamma_{C_2'}
\end{split}
\end{equation*}

§.§.§ Vertex Number 4

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) – (2.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:360.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (1.000000, 0.000000) arc (180.000000:0.000000:1.250000);
[] (3.500000, 0.000000) arc (0.000000:-180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\left[\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_4}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1} , \Gamma_{C_2} \right]=e
\end{split}
\end{equation*}

§.§.§ Vertex Number 5

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) – (4.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) – (4.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) arc (0.000000:180.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) arc (0.000000:180.000000:2.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (5.000000, 0.000000) arc (0.000000:180.000000:2.500000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\left\{\Gamma_{L_4}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{C_1'}^{-1}\Gamma_{C_2'}\Gamma_{C_1'}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_4}  , \Gamma_{L_2} \right\}=e
\end{split}
\end{equation*}

§.§.§ Vertex Number 6

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) – (5.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:0.000000:1.500000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:2.500000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (0.000000, 0.000000) arc (180.000000:0.000000:2.750000);
[] (5.500000, 0.000000) arc (0.000000:-180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\left[\Gamma_{C_2'}\Gamma_{C_1'}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_4}\Gamma_{L_2}\Gamma_{L_4}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{C_1'}^{-1}\Gamma_{C_2'}^{-1} , \Gamma_{C_1'} \right]=e
\end{split}
\end{equation*}

§.§.§ Vertex Number 7

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) – (1.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 5, 6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) – (1.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) – (1.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:360.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:360.000000:1.500000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) – (3.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (3.000000, 0.000000) – (2.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\left\{\Gamma_{C_1}  , \Gamma_{C_2} \right\}=e
\end{split}
\end{equation*}

§.§.§ Vertex Number 8

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) – (4.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 1, 2 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) – (4.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 5, 6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (180.000000:360.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) – (5.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) – (5.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (5.000000, 0.000000) – (4.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\left\{\Gamma_{C_2'}  , \Gamma_{C_1'} \right\}=e
\end{split}
\end{equation*}

§.§.§ Vertex Number 9

[scale=1 , xshift=-100]
[] (5.000000, 0.000000) – (6.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (5.000000, 0.000000) – (6.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 1, 2 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (5.000000, 0.000000) – (6.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 5, 6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:1.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:1.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:0.000000:2.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:3.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (0.000000, 0.000000) arc (180.000000:0.000000:3.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\left[\Gamma_{C_2'}\Gamma_{C_1'}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_4}\Gamma_{L_2}\Gamma_{L_4}^{-1}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}\Gamma_{C_1'}^{-1}\Gamma_{C_2'}^{-1} , \Gamma_{L_1} \right]=e
\end{split}
\end{equation*}

§.§.§ Vertex Number 10

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) – (3.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 6, 7 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) – (3.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:360.000000:1.250000);
[] (4.500000, 0.000000) arc (0.000000:180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 1, 2 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:360.000000:1.250000);
[] (4.500000, 0.000000) arc (0.000000:180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 5, 6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:360.000000:1.750000);
[] (5.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:-180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:360.000000:1.750000);
[] (5.500000, 0.000000) arc (0.000000:180.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:1.500000);
[] (5.500000, 0.000000) arc (0.000000:180.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:360.000000:0.750000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (3.500000, 0.000000) arc (180.000000:360.000000:1.000000);
[] (5.500000, 0.000000) arc (0.000000:180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:0.000000:1.250000);
[] (3.500000, 0.000000) arc (180.000000:360.000000:1.000000);
[] (5.500000, 0.000000) arc (0.000000:180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (1.000000, 0.000000) arc (180.000000:0.000000:1.250000);
[] (3.500000, 0.000000) arc (180.000000:360.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\left[\Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_4}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1} , \Gamma_{C_2'} \right]=e
\end{split}
\end{equation*}

§.§.§ Vertex Number 11

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) – (4.000000, 0.000000) ;
[] (4.000000, 0.000000) – (5.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 3, 4 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (4.000000, 0.000000) – (5.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 6, 7 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (4.000000, 0.000000) arc (180.000000:360.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:360.000000:1.000000);
[] (4.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (4.500000, 0.000000) arc (180.000000:360.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 1, 2 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:360.000000:1.000000);
[] (4.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (4.500000, 0.000000) arc (180.000000:360.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 5, 6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:360.000000:1.500000);
[] (5.000000, 0.000000) arc (0.000000:-180.000000:1.250000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (3.500000, 0.000000) arc (180.000000:360.000000:0.500000);
[] (4.500000, 0.000000) arc (180.000000:0.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:360.000000:1.500000);
[] (5.000000, 0.000000) arc (0.000000:180.000000:0.750000);
[] (3.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:1.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:1.250000);
[] (5.000000, 0.000000) arc (0.000000:180.000000:0.750000);
[] (3.500000, 0.000000) arc (0.000000:-180.000000:0.400000);
[] (2.700000, 0.000000) arc (180.000000:0.000000:1.650000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:360.000000:0.750000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (3.500000, 0.000000) arc (180.000000:360.000000:0.750000);
[] (5.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:0.000000:1.250000);
[] (3.500000, 0.000000) arc (180.000000:360.000000:0.750000);
[] (5.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (1.000000, 0.000000) arc (180.000000:0.000000:1.500000);
[] (4.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\left( \Gamma_{C_2'}^{-1}\Gamma_{L_1}\Gamma_{C_2'}  \right) \left( \Gamma_{C_1'}  \right) \left( \Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_4}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}  \right)& = \\
\left( \Gamma_{C_1'}  \right) \left( \Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_4}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}  \right) \left( \Gamma_{C_2'}^{-1}\Gamma_{L_1}\Gamma_{C_2'}  \right)& = \\
\left( \Gamma_{C_1}\Gamma_{C_2}\Gamma_{L_4}\Gamma_{C_2}^{-1}\Gamma_{C_1}^{-1}  \right) \left( \Gamma_{C_2'}^{-1}\Gamma_{L_1}\Gamma_{C_2'}  \right) \left( \Gamma_{C_1'}  \right)
\end{split}
\end{equation*}

§.§.§ Vertex Number 12

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) – (3.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 4,5,6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (180.000000:360.000000:1.500000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 3, 4 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (180.000000:360.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 6, 7 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (180.000000:360.000000:1.500000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (4.500000, 0.000000) arc (180.000000:360.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 1, 2 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (4.500000, 0.000000) arc (180.000000:360.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 5, 6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (180.000000:360.000000:0.750000);
[] (4.500000, 0.000000) arc (180.000000:0.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (180.000000:0.000000:1.500000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (180.000000:0.000000:1.500000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (4.000000, 0.000000) arc (180.000000:0.000000:1.000000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (5.000000, 0.000000) – (6.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\left\{\Gamma_{C_2'}  , \Gamma_{L_1} \right\}=e
\end{split}
\end{equation*}

§.§.§ Vertex Number 13

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) – (2.000000, 0.000000) ;
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 3, 4 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:360.000000:1.250000);
[] (3.500000, 0.000000) arc (0.000000:180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 4,5,6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:360.000000:2.250000);
[] (5.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:-180.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 3, 4 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:360.000000:2.250000);
[] (5.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:-180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 6, 7 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:360.000000:2.750000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (5.500000, 0.000000) arc (0.000000:-180.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (180.000000:360.000000:2.750000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (5.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 1, 2 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:2.500000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (5.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 5, 6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:2.500000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.000000);
[] (4.500000, 0.000000) arc (0.000000:-180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (1.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:2.500000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 2, 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (2.000000, 0.000000) arc (0.000000:-180.000000:1.250000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (0.500000, 0.000000) arc (180.000000:360.000000:0.500000);
[] (1.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:2.000000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{\mathbb{R}I_{2}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (0.000000:-180.000000:1.750000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (0.500000, 0.000000) arc (180.000000:360.000000:1.000000);
[] (2.500000, 0.000000) arc (180.000000:0.000000:0.500000);
[] (3.500000, 0.000000) arc (180.000000:360.000000:1.500000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 1,2,3 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (3.000000, 0.000000) arc (0.000000:-180.000000:0.750000);
[] (1.500000, 0.000000) arc (180.000000:0.000000:1.000000);
[] (3.500000, 0.000000) arc (180.000000:360.000000:1.500000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}I_{4}}^{1/2}\langle 3 \rangle $  (6.000000, 0.000000) ;

[] (2.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (3.500000, 0.000000) arc (180.000000:360.000000:1.500000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) $ L_2 $;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) $ L_4 $;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) $ C_2 $;
[draw, rectangle, color=red, fill=white] (vert_3) at (3.000000,0.000000) $ C_1 $;
[draw, rectangle, color=red, fill=white] (vert_4) at (4.000000,0.000000) $ C_1' $;
[draw, rectangle, color=red, fill=white] (vert_5) at (5.000000,0.000000) $ C_2' $;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) $ L_1 $;

\begin{equation*}\begin{split}
\Gamma_{L_1}^{-1}\Gamma_{C_1}\Gamma_{C_2}\Gamma_{C_1}^{-1}\Gamma_{L_1}  = \Gamma_{C_2'}
\end{split}
\end{equation*}

§.§.§ Vertex Number 14

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, rectangle, color=red, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta_{I_{2}\mathbb{R}}^{1/2}\langle 2 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:0.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 3, 4 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:1.000000);
[] (3.500000, 0.000000) arc (0.000000:180.000000:0.900000);
[] (1.700000, 0.000000) arc (180.000000:360.000000:0.650000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta \langle 4,5,6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:2.000000);
[] (5.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:-180.000000:1.000000);
[] (2.500000, 0.000000) arc (0.000000:180.000000:0.400000);
[] (1.700000, 0.000000) arc (180.000000:360.000000:1.650000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 3, 4 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:2.000000);
[] (5.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (3.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:1.250000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 6, 7 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:2.500000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (5.500000, 0.000000) arc (0.000000:-180.000000:1.000000);
[] (3.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:1.750000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 4, 5 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (180.000000:360.000000:2.500000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (5.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:180.000000:1.000000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:0.500000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:0.400000);
[] (4.300000, 0.000000) arc (180.000000:360.000000:0.850000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta^{2}\langle 1, 2 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (0.000000:-180.000000:1.000000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:1.100000);
[] (1.700000, 0.000000) arc (180.000000:360.000000:2.400000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (5.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (4.500000, 0.000000) arc (0.000000:180.000000:1.000000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:0.500000);
[] (3.500000, 0.000000) arc (180.000000:0.000000:0.400000);
[] (4.300000, 0.000000) arc (180.000000:360.000000:0.850000);
[draw, circle, color=black, fill=white] (vert_0) at (0.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_1) at (1.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_2) at (2.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_3) at (3.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_4) at (4.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_5) at (5.000000,0.000000) ;
[draw, circle, color=black, fill=white] (vert_6) at (6.000000,0.000000) ;

[scale=1 , xshift=100]
[->] (0.000000, 0.000000) – node [above, midway] $ \Delta\langle 5, 6 \rangle $  (6.000000, 0.000000) ;

[scale=1 , xshift=-100]
[] (0.000000, 0.000000) arc (180.000000:0.000000:0.750000);
[] (1.500000, 0.000000) arc (0.000000:-180.000000:1.000000);
[] (-0.500000, 0.000000) arc (180.000000:0.000000:1.100000);
[] (1.700000, 0.000000) arc (180.000000:360.000000:2.400000);
[] (6.500000, 0.000000) arc (0.000000:180.000000:1.000000);
[] (4.500000, 0.000000) arc (0.000000:-180.000000:0.500000);
[] (3.500000, 0.000000) arc (0.000000:180.000000:0.500000);
[] (2.500000, 0.000000) arc (180.000000:360.000000:1.100000);