[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (190,35.09) .. controls (194.93,40.11) and (194.93,40.11) .. (195.02,45.05) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (160,35.12) .. controls (155.15,30.08) and (155.15,30.08) .. (155.08,25.14) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (180,15.17) .. controls (184.92,10.14) and (185,10.18) .. (185,5.2) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (170,15.17) .. controls (165.08,10.14) and (165.08,10.14) .. (165,5.2) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (174.97,20.16) – (190,35.09) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (155.02,125) – (155,104.95) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (155.07,45.13) .. controls (155.03,40.07) and (154.99,40.1) .. (160,35.12) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (174.97,20.16) – (160,35.16) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (175,20.16) – (170,15.17) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (175,20.16) – (180,15.17) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (175,40.07) – (175,20.16) ;
[shift=(175,40.07), rotate = 270] [color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 0; blue, 0   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (155.08,25.14) – (155.06,5.09) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (194.98,25.1) – (194.95,5.05) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (230.05,35.36) .. controls (233.77,40.17) and (233.77,39.67) .. (235.05,45.44) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (245.02,20.16) – (245,5.1) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (230.02,94.98) .. controls (225.11,100) and (225.11,100) .. (225.02,104.95) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (260.02,94.98) .. controls (264.94,100) and (264.94,100) .. (265.02,104.95) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (235.1,85) – (240.09,65.26) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (235.05,45.44) – (240.11,65.43) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (260.02,35.09) .. controls (264.93,30.05) and (264.93,30.05) .. (265,25.1) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (250.05,65.04) – (251.72,71.59) – (251.72,71.59) – (255.02,84.89) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (265.05,125) – (265.02,104.95) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (230.02,94.98) .. controls (225.11,89.95) and (225.11,89.95) .. (225.02,85) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (260.02,94.98) .. controls (264.98,89.92) and (264.98,89.92) .. (265.04,84.97) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (225.02,85) – (225.09,45.13) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (265.04,84.97) – (265.05,45.05) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (260.02,35.09) .. controls (264.95,40.11) and (264.95,40.11) .. (265.05,45.05) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (230.02,35.12) .. controls (225.18,30.08) and (225.18,30.08) .. (225.11,25.14) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (250.02,15.17) .. controls (254.94,10.14) and (255.02,10.18) .. (255.02,5.2) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (240.02,15.17) .. controls (235.11,10.14) and (235.11,10.14) .. (235.02,5.2) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (244.99,20.16) – (260.02,35.09) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (240.09,65.26) – (250.05,65.04) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (250.05,65.04) – (254.98,45) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (225.05,125) – (225.02,104.95) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (225.09,45.13) .. controls (225.05,40.07) and (225.01,40.1) .. (230.02,35.12) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (244.99,20.16) – (230.02,35.16) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (245.02,20.16) – (240.02,15.17) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (245.02,20.16) – (250.02,15.17) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (245.02,40.07) – (245.02,20.16) ;
[shift=(245.02,40.07), rotate = 270] [color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 0; blue, 0   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (225.11,25.14) – (225.08,5.09) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (265,25.1) – (264.98,5.05) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (260.02,94.98) .. controls (244.77,105.17) and (245.27,105.17) .. (230.02,94.98) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (165.1,84.92) – (170.09,65.17) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (165.05,45.36) – (170.11,65.35) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (180.05,64.95) – (181.72,71.51) – (181.72,71.51) – (185.02,84.81) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (170.09,65.17) – (180.05,64.95) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (180.05,64.95) – (184.98,44.92) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ][line width=0.75]  (95,85) – (115,85) – (115,45) – (95,45) – cycle ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (385.02,20.16) – (385,5.1) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (390.02,15.17) .. controls (394.94,10.14) and (395.02,10.18) .. (395.02,5.2) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (380.02,15.17) .. controls (375.11,10.14) and (375.11,10.14) .. (375.02,5.2) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (384.99,20.16) – (395.35,30) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (384.99,20.16) – (375.35,30.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (385.02,20.16) – (380.02,15.17) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (385.02,20.16) – (390.02,15.17) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (385.02,40.07) – (385.02,20.16) ;
[shift=(385.02,40.07), rotate = 270] [color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 0; blue, 0   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (365,44.92) – (365.08,5.09) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (405,45) – (404.98,5.05) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (370.05,55.33) .. controls (366.32,50.52) and (366.27,50.69) .. (365,44.92) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (400.02,55.06) .. controls (404.93,50.02) and (404.93,50.02) .. (405,45.07) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (372,43.21) – (372.06,36.66) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (398.67,43.21) – (398.7,36.61) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (395.35,30) .. controls (398.63,33.33) and (398.63,33.33) .. (398.7,36.61) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (372.06,36.66) .. controls (372.03,33.3) and (372.01,33.32) .. (375.35,30.02) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (375.33,49.84) .. controls (372.06,46.51) and (372.06,46.51) .. (372,43.23) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (395.33,49.84) .. controls (398.64,46.49) and (398.64,46.49) .. (398.68,43.21) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (395.33,49.84) .. controls (385.17,56.6) and (385.5,56.6) .. (375.33,49.84) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (400.02,114.94) .. controls (396.3,110.13) and (396.3,110.63) .. (395.02,104.86) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (370.1,115.05) .. controls (373.83,110.24) and (373.83,110.74) .. (375.1,104.97) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (400.02,55.06) .. controls (396.3,59.86) and (396.3,59.36) .. (395.02,65.14) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (370.05,55.33) .. controls (373.77,60.13) and (373.77,59.63) .. (375.05,65.41) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (370,115.02) .. controls (365.08,120.05) and (365.08,120.05) .. (365,125) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (400,115.02) .. controls (404.92,120.05) and (404.92,120.05) .. (405,125) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (375.1,104.97) – (380.09,85.22) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (375.05,65.41) – (380.11,85.4) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (390.05,85) – (391.72,91.56) – (391.72,91.56) – (395.02,104.86) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (380.09,85.22) – (390.05,85) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (390.05,85) – (394.98,64.97) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (315.02,20.16) – (315,5.1) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (320.02,15.17) .. controls (324.94,10.14) and (325.02,10.18) .. (325.02,5.2) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (310.02,15.17) .. controls (305.11,10.14) and (305.11,10.14) .. (305.02,5.2) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (314.99,20.16) – (325.35,30) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (314.99,20.16) – (305.35,30.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (315.02,20.16) – (310.02,15.17) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (315.02,20.16) – (320.02,15.17) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (315.02,40.07) – (315.02,20.16) ;
[shift=(315.02,40.07), rotate = 270] [color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 0; blue, 0   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (295,44.92) – (295.08,5.09) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (335,45) – (334.98,5.05) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (300.05,55.33) .. controls (296.32,50.52) and (296.27,50.69) .. (295,44.92) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (330.02,55.06) .. controls (334.93,50.02) and (334.93,50.02) .. (335,45.07) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (302,43.21) – (302.06,36.66) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (328.67,43.21) – (328.7,36.61) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (325.35,30) .. controls (328.63,33.33) and (328.63,33.33) .. (328.7,36.61) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (302.06,36.66) .. controls (302.03,33.3) and (302.01,33.32) .. (305.35,30.02) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (305.33,49.84) .. controls (302.06,46.51) and (302.06,46.51) .. (302,43.23) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (325.33,49.84) .. controls (328.64,46.49) and (328.64,46.49) .. (328.68,43.21) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (325.33,49.84) .. controls (315.17,56.6) and (315.5,56.6) .. (305.33,49.84) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (330.02,114.94) .. controls (326.3,110.13) and (326.3,110.63) .. (325.02,104.86) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (300.1,115.05) .. controls (303.83,110.24) and (303.83,110.74) .. (305.1,104.97) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (330.02,55.06) .. controls (326.3,59.86) and (326.3,59.36) .. (325.02,65.14) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (300.05,55.33) .. controls (303.77,60.13) and (303.77,59.63) .. (305.05,65.41) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (300,115.02) .. controls (295.08,120.05) and (295.08,120.05) .. (295,125) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (330,115.02) .. controls (334.92,120.05) and (334.92,120.05) .. (335,125) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (305.1,104.97) – (310.09,85.22) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (305.05,65.41) – (310.11,85.4) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (320.05,85) – (321.72,91.56) – (321.72,91.56) – (325.02,104.86) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (310.09,85.22) – (320.05,85) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (320.05,85) – (324.98,64.97) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ][line width=0.75]  (150,115) – (200,115) – (200,30) – (150,30) – cycle ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ][line width=0.75]  (220,115) – (270,115) – (270,30) – (220,30) – cycle ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ][line width=0.75]  (300.05,55.33) – (330,55.33) – (330,15) – (300.05,15) – cycle ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ][line width=0.75]  (370.05,55.33) – (400,55.33) – (400,15) – (370.05,15) – cycle ;

(149,72.6) node [anchor=north east][inner sep=0.75pt]  [rotate=-180,xscale=-1]  $=$;

(74,72.6) node [anchor=north east][inner sep=0.75pt]  [rotate=-180,xscale=-1]  $+$;

(29,128.4) node [anchor=north west][inner sep=0.75pt]    $5)$;

(99,128.4) node [anchor=north west][inner sep=0.75pt]    $6)$;

(168.03,127.4) node [anchor=north west][inner sep=0.75pt]    $7)$;

(239,128.4) node [anchor=north west][inner sep=0.75pt]    $8)$;

(309.97,128.4) node [anchor=north west][inner sep=0.75pt]    $9)$;

(379,127.4) node [anchor=north west][inner sep=0.75pt]    $10)$;

(214.02,72.6) node [anchor=north east][inner sep=0.75pt]  [rotate=-180,xscale=-1]  $+$;

(354.02,72.6) node [anchor=north east][inner sep=0.75pt]  [rotate=-180,xscale=-1]  $+$;

(289,72.6) node [anchor=north east][inner sep=0.75pt]  [rotate=-180,xscale=-1]  $=$;

From $5)$ and $6)$, to $7)$ and $8)$ we apply Eq. 2.15. From $7)$ and $8)$ to $9)$ and $10)$, we repeatedly apply Eq. 2.20 and Eq. 2.22. As in Figure <ref>,
we recognize in $9)$ and $10)$ compositions equivalent to the zero morphism.

The Forking Path Conjecture is true for $w_{0,4}$.

By <ref> and <ref>, we have that any path starting at $\mathbf{s}$ and ending in  $\mathbf{t}$, that does not visit $\mathbf{s}$ or $\mathbf{t}$ in the rest of the path, and that is located on the left half of the Zamolodchikov cycle (See Figure <ref>) is equivalent to a direct path. The same result is true if one considers paths from $\mathbf{s}$ to $\mathbf{t}$ or from $\mathbf{t}$ to $\mathbf{s}$, located on the left  or on the right half of the cycle.

The reason for this is that the proofs for the four cases will be the same as before, but turning the diagrams upside down for the case $\mathbf{t}$ to $\mathbf{s}$ in the left half, applying a vertical axial symmetry to the diagrams  for the right half for the case $\mathbf{s}$ to $\mathbf{t}$ and turning the diagrams upside down and applying a vertical axial symmetry to the diagrams for the case $\mathbf{t}$ to $\mathbf{s}$ on the right[ A deeper reason for this  is that we are implicitly applying some equivalences of categories. There  is a contravariant equivalence of monoidal categories $\mathbb{S}Bim\rightarrow \mathbb{S}Bim$, given by the flip (that sends a diagram to its horizontal flip) and  also an auto-equivalence of $\mathbb{S}Bim$  associated to the only non-trivial automorphism of the Dynkin diagram of type $A_n$, sending $s_i\mapsto s_{n-i+1}.$ ].

So we conclude that any complete path in $\Gamma_{w_{0,4}}$ has  a subpath that is equivalent to a direct path.
Therefore, by Proposition <ref> the proof of the FPC for $w_0$ in $S_4$ is complete.

Now we verify the conjecture for the remaining elements in $S_4$ different from $12321$. The elements $w$ and their $\Gamma_w$ graphs oriented according to Manin-Schechtman  are given in the following table.

e     1     2     21     12     121

$\bullet$     $\bullet$     $\bullet$     $\bullet$     $\bullet$     $\bullet \rightarrow \bullet$

3   31   32   321   312   3121

$\bullet$     $\bullet$     $\bullet$     $\bullet$     $\bullet$     $\bullet \rightarrow \bullet$

23   231   232   2321   2312   23121

$\bullet$     $\bullet$    $\bullet \rightarrow \bullet$     $\bullet \rightarrow \bullet$      $\bullet$     $\bullet \rightarrow \bullet \rightarrow \bullet$

123   1231   1232   12321   12312   123121

$\bullet$     $\bullet \rightarrow \bullet$      $\bullet \rightarrow \bullet$      $\bullet \rightarrow \bullet \rightarrow \bullet$      $\bullet \rightarrow \bullet \rightarrow \bullet$      $\mathrm{Zam}$

Elements in $S_4$ and their conflated expression graphs.

The last entry $\mathrm{Zam}$ is the Zamolodchikov cycle, as introduced in Example <ref>. There is no need to verify the trivial graphs $\Gamma_w$ ($\bullet$), since the only possible morphism is the $\mathrm{id}$. For any $\bullet \rightarrow \bullet$ case, the proof of the Forking Path Conjecture follows easily from Lemma <ref>. It remains to check the $\bullet \rightarrow \bullet \rightarrow \bullet$ cases. We will concentrate in the cases  23121 and 12312 because the other case is the one giving the counterexample to the Forking Path Conjecture. We now study the element 23121.

The Forking path conjecture is true for the element 23121.

In this case we can speak of a simplified path $p$ similar to that of Definition <ref>. Consider $\{x,y\}=\{\mathbf{s}, \mathbf{t}\}$. These paths will be of the form
$p_a\rightarrow x\rightarrow y\rightarrow p_z$, where $p_a$ could be equal to $x$, and $p_z$ could be equal to $y$, or alternatively, of the form
$p_a\rightarrow x\rightarrow y\rightarrow x\rightarrow p_z$, where  $p_a$ and $p_z$ could be equal to $x$. The path $p_a \rightarrow x$ (resp. $y\rightarrow p_z$ in the first case, and $x\rightarrow p_z$ in the second) will have length one only when $p_a=c$ (resp. $p_z=c$), where $c$ is the only vertex different to $\mathbf{s}$ and  $\mathbf{t}$. We consider simplified paths $p$ and $q$.

$\bullet$ We first study the case $p_a, p_z\in \{\mathbf{s}, \mathbf{t}\}$. It is immediate that $p\simeq q$, since there is only one possible path, i.e., $p=q$.

$\bullet$ Consider the case $p_a=\mathbf{s}$ and $p_z=c$. There are two possible simplified paths, $P_1:=[\mathbf{s}, c, \mathbf{t} , c]$ and $P_2:=[\mathbf{s}, c, \mathbf{t} , c, \mathbf{s} , c].$
We will prove that $P_1$ is equivalent to $P_2$. In the following figure,  part 1) represents the path morphism of $P_2$.

every picture/.style=line width=0.75pt

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[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (130.03,200) .. controls (129.95,210.15) and (129.88,210.08) .. (139.9,220.07) ;

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[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (150.03,230) – (140.03,240) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (150.03,230) – (160.04,240) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (200.01,280) .. controls (209.87,269.94) and (209.87,269.94) .. (210.05,260.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (170.01,260) .. controls (169.93,270.15) and (169.85,270.08) .. (179.87,280.07) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (200.01,300) .. controls (209.84,310.08) and (210.01,310) .. (210.01,320) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (180.01,300) .. controls (170.17,310.08) and (170.17,310.08) .. (170.01,320) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (190.01,290) – (180.01,300) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (190.01,290) – (200.01,300) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (189.94,290) – (200.01,280) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (179.87,280.07) – (188.33,288.52) – (190.01,290) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (130.03,260) – (130.01,320) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (210.01,20) – (210.01,260) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (130.03,20) – (130.01,140) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (189.94,290) – (189.94,320) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (170.01,20) – (169.98,140) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (300.01,30) – (300.01,20) ;
[shift=(300.01,30), rotate = 270] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (300.01,130) – (300.01,140) ;
[shift=(300.01,130), rotate = 90] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[line width=0.75]  (235.01,140) – (285,140) – (285,260) – (235.01,260) – cycle ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (520.01,140) – (520.01,290) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (410.01,20) – (410.01,290) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (370.03,230) – (370.01,320) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (359.9,220.07) – (370.03,230) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (369.97,230) – (380.04,220) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (370.05,20) – (370.01,170) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (480,20) – (479.95,320) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (529.95,280) .. controls (539.81,269.94) and (539.81,269.94) .. (540,260.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (499.95,260) .. controls (499.87,270.15) and (499.8,270.08) .. (509.82,280.07) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (529.95,300) .. controls (539.79,310.08) and (539.95,310) .. (539.95,320) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (509.95,300) .. controls (500.12,310.08) and (500.12,310.08) .. (499.95,320) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (519.95,290) – (509.95,300) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (519.95,290) – (529.95,300) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (519.89,290) – (529.95,280) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (509.82,280.07) – (518.28,288.52) – (519.95,290) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (459.98,260) – (459.95,320) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (539.95,140) – (539.95,260) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (529.95,40) .. controls (539.81,29.94) and (539.81,29.94) .. (540,20.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (499.95,20) .. controls (499.87,30.15) and (499.8,30.08) .. (509.82,40.07) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (519.95,50) – (519.9,110) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (519.89,50) – (529.95,40) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (509.82,40.07) – (518.28,48.52) – (519.95,50) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (529.97,120) .. controls (539.8,130.08) and (539.97,130) .. (539.97,140) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (509.97,120) .. controls (500.13,130.08) and (500.13,130.08) .. (499.97,140) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (519.97,110) – (509.97,120) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (519.97,110) – (529.97,120) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (459.98,20) – (459.95,140) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (519.89,290) – (519.89,320) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (380.02,160) .. controls (389.88,149.94) and (389.88,149.94) .. (390.07,140.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (350.02,140) .. controls (349.94,150.15) and (349.86,150.08) .. (359.88,160.07) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (380.02,180) .. controls (389.85,190.08) and (390.02,190) .. (390.02,200) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (360.02,180) .. controls (350.19,190.08) and (350.19,190.08) .. (350.02,200) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (370.02,170) – (360.02,180) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (370.02,170) – (380.02,180) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (369.95,170) – (380.02,160) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (359.88,160.07) – (368.34,168.52) – (370.02,170) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (370.02,170) – (370.03,230) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (380.04,220) .. controls (389.89,209.94) and (389.89,209.94) .. (390.08,200.02) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (350.03,200) .. controls (349.95,210.15) and (349.88,210.08) .. (359.9,220.07) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (380.04,240) .. controls (389.87,250.08) and (390.04,250) .. (390.04,260) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (360.03,240) .. controls (350.2,250.08) and (350.2,250.08) .. (350.03,260) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (370.03,230) – (360.03,240) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (370.03,230) – (380.04,240) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (420.01,280) .. controls (429.87,269.94) and (429.87,269.94) .. (430.05,260.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (390.01,260) .. controls (389.93,270.15) and (389.85,270.08) .. (399.87,280.07) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (420.01,300) .. controls (429.84,310.08) and (430.01,310) .. (430.01,320) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (400.01,300) .. controls (390.17,310.08) and (390.17,310.08) .. (390.01,320) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (410.01,290) – (400.01,300) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (410.01,290) – (420.01,300) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (409.94,290) – (420.01,280) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (399.87,280.07) – (408.33,288.52) – (410.01,290) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (350.03,260) – (350.01,320) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (430.01,20) – (430.01,260) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (350.03,20) – (350.01,140) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (409.94,290) – (409.94,320) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (390.01,20) – (389.98,140) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (520.01,30) – (520.01,20) ;
[shift=(520.01,30), rotate = 270] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (520.01,130) – (520.01,140) ;
[shift=(520.01,130), rotate = 90] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (630,140) – (630,290) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (590,20) – (590,145) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (599.96,160) .. controls (609.81,149.94) and (609.81,149.94) .. (610,140.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (569.95,140) .. controls (569.87,150.15) and (569.8,150.08) .. (579.82,160.07) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (589.89,170) – (599.96,160) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (579.82,160.07) – (588.28,168.52) – (589.96,170) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (589.96,170) – (589.97,230) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (599.97,240) .. controls (609.8,250.08) and (609.97,250) .. (609.97,260) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (579.97,240) .. controls (570.14,250.08) and (570.14,250.08) .. (569.97,260) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (589.97,230) – (579.97,240) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (589.97,230) – (599.97,240) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (590,260) – (589.94,320) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (639.94,280) .. controls (649.8,269.94) and (649.8,269.94) .. (649.99,260.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (609.94,260) .. controls (609.86,270.15) and (609.79,270.08) .. (619.81,280.07) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (639.94,300) .. controls (649.78,310.08) and (649.94,310) .. (649.94,320) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (619.94,300) .. controls (610.11,310.08) and (610.11,310.08) .. (609.94,320) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (629.94,290) – (619.94,300) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (629.94,290) – (639.94,300) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (629.88,290) – (639.94,280) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (619.81,280.07) – (628.27,288.52) – (629.94,290) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (569.97,260) – (569.94,320) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (649.94,140) – (649.94,260) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (639.94,40) .. controls (649.8,29.94) and (649.8,29.94) .. (649.99,20.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (609.94,20) .. controls (609.86,30.15) and (609.79,30.08) .. (619.81,40.07) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (629.94,50) – (629.89,110) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (629.88,50) – (639.94,40) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (619.81,40.07) – (628.27,48.52) – (629.94,50) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (639.96,120) .. controls (649.79,130.08) and (649.96,130) .. (649.96,140) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (619.96,120) .. controls (610.12,130.08) and (610.12,130.08) .. (609.96,140) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (629.96,110) – (619.96,120) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (629.96,110) – (639.96,120) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (569.97,20) – (569.94,140) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (629.88,290) – (629.88,320) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (630,30) – (630,20) ;
[shift=(630,30), rotate = 270] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (630,130) – (630,140) ;
[shift=(630,130), rotate = 90] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (499.97,140) – (499.95,260) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (459.95,140) – (460,265) ;

[line width=0.75]  (495,100) – (544.99,100) – (544.99,305) – (495,305) – cycle ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (590,155) – (590,145) ;
[shift=(590,155), rotate = 270] [color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][fill=rgb, 255:red, 126; green, 211; blue, 33   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (590,245) – (590,260) ;
[shift=(590,245), rotate = 90] [color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][fill=rgb, 255:red, 126; green, 211; blue, 33   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;
[line width=0.75]  (565,100) – (655,100) – (655,305) – (565,305) – cycle ;

[line width=0.75]  (54.98,20) – (104.97,20) – (104.97,140) – (54.98,140) – cycle ;

(108,176.5) node [anchor=north west][inner sep=0.75pt]   [align=left] =;

(218.01,177) node [anchor=north west][inner sep=0.75pt]   [align=left] +;

(328,176.5) node [anchor=north west][inner sep=0.75pt]   [align=left] =;

(438.01,177) node [anchor=north west][inner sep=0.75pt]   [align=left] +;

(546,177) node [anchor=north west][inner sep=0.75pt]   [align=left] +;

(51,327.4) node [anchor=north west][inner sep=0.75pt]    $1)$;

(161,327.4) node [anchor=north west][inner sep=0.75pt]    $2)$;

(274,327.4) node [anchor=north west][inner sep=0.75pt]    $3)$;

(384,327.4) node [anchor=north west][inner sep=0.75pt]    $4)$;

(494,327.4) node [anchor=north west][inner sep=0.75pt]    $5)$;

(604,327.4) node [anchor=north west][inner sep=0.75pt]    $6)$;

We begin in $1)$ applying the relation seen in Figure <ref>. Then, we apply the same relation in $3)$. Diagram $5)$ is zero by the same reason as diagram $9)$ in Figure <ref>. In $6)$ we focus on the red and blue strands because we can retract the green dots. We have the following diagrams.

every picture/.style=line width=0.75pt

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (80,190) – (80,215) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (49.96,85) .. controls (59.81,74.94) and (59.81,74.94) .. (60,65.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (19.95,65) .. controls (19.87,75.15) and (19.8,75.08) .. (29.82,85.07) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (39.89,95) – (49.96,85) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (29.82,85.07) – (38.28,93.52) – (39.96,95) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (39.96,95) – (39.97,155) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (49.97,165) .. controls (59.8,175.08) and (59.97,175) .. (59.97,185) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (29.97,165) .. controls (20.14,175.08) and (20.14,175.08) .. (19.97,185) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (39.97,155) – (29.97,165) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (39.97,155) – (49.97,165) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (89.94,205) .. controls (99.8,194.94) and (99.8,194.94) .. (99.99,185.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (59.94,185) .. controls (59.86,195.15) and (59.79,195.08) .. (69.81,205.07) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (89.94,225) .. controls (99.78,235.08) and (99.94,235) .. (99.94,245) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (69.94,225) .. controls (60.11,235.08) and (60.11,235.08) .. (59.94,245) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (79.94,215) – (69.94,225) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (79.94,215) – (89.94,225) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (79.88,215) – (89.94,205) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (69.81,205.07) – (78.27,213.52) – (79.94,215) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (19.97,185) – (19.94,245) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (99.94,65) – (99.94,185) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (79.99,10) – (79.89,35) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (89.96,45) .. controls (99.79,55.08) and (99.96,55) .. (99.96,65) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (69.96,45) .. controls (60.12,55.08) and (60.12,55.08) .. (59.96,65) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (79.96,35) – (69.96,45) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (79.96,35) – (89.96,45) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (19.99,10) – (19.94,65) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (79.88,215) – (79.88,245) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (80,180) – (80,190) ;
[shift=(80,180), rotate = 90] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[line width=0.75]  (55,140) – (105,140) – (105,230) – (55,230) – cycle ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (224.93,205) .. controls (234.79,194.94) and (234.79,194.94) .. (234.98,185.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (194.93,185) .. controls (194.85,195.15) and (194.78,195.08) .. (204.8,205.07) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (214.87,215) – (224.93,205) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (204.8,205.07) – (213.26,213.52) – (214.93,215) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (154.96,185) – (154.93,245) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (234.93,65) – (234.93,185) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (214.99,10) – (214.88,35) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (224.95,45) .. controls (234.78,55.08) and (234.95,55) .. (234.95,65) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (204.95,45) .. controls (195.11,55.08) and (195.11,55.08) .. (194.95,65) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (214.95,35) – (204.95,45) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (214.95,35) – (224.95,45) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (154.99,10) – (154.93,65) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (214.87,215) – (214.87,245) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (184.95,85) .. controls (194.81,74.94) and (194.81,74.94) .. (195,65.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (154.95,65) .. controls (154.87,75.15) and (154.8,75.08) .. (164.82,85.07) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (174.89,95) – (184.95,85) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (164.82,85.07) – (173.28,93.52) – (174.95,95) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (174.95,95) – (174.97,155) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (184.97,165) .. controls (194.8,175.08) and (194.97,175) .. (194.97,185) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (164.97,165) .. controls (155.13,175.08) and (155.13,175.08) .. (154.97,185) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (174.97,155) – (164.97,165) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (174.97,155) – (184.97,165) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (235,235) – (235,245) ;
[shift=(235,235), rotate = 90] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (195,235) – (195,245) ;
[shift=(195,235), rotate = 90] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (294.98,185) – (294.95,245) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (374.95,65) – (374.95,185) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (355.01,10) – (354.9,35) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (364.97,45) .. controls (374.8,55.08) and (374.97,55) .. (374.97,65) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (344.97,45) .. controls (335.13,55.08) and (335.13,55.08) .. (334.97,65) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (354.97,35) – (344.97,45) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (354.97,35) – (364.97,45) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (295.01,10) – (294.95,65) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (324.97,85) .. controls (334.83,74.94) and (334.83,74.94) .. (335.02,65.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (294.97,65) .. controls (294.89,75.15) and (294.82,75.08) .. (304.84,85.07) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (314.91,95) – (324.97,85) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (304.84,85.07) – (313.3,93.52) – (314.97,95) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (314.97,95) – (314.99,155) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (324.99,165) .. controls (334.82,175.08) and (334.99,175) .. (334.99,185) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (304.99,165) .. controls (295.15,175.08) and (295.15,175.08) .. (294.99,185) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (314.99,155) – (304.99,165) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (314.99,155) – (324.99,165) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (355,235) – (355,245) ;
[shift=(355,235), rotate = 90] [color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 0; blue, 0   ,fill opacity=1 ][line width=1.5]      (0, 0) circle [x radius= 2.61, y radius= 2.61]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (345.09,225.11) .. controls (335.28,234.08) and (335.14,234.04) .. (335,242.84) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (365.1,225.07) .. controls (374.95,234) and (374.81,234.04) .. (375,242.84) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (345.09,225.11) .. controls (355.05,218.3) and (355.05,218.3) .. (365.1,225.07) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (345.09,199.89) .. controls (335.28,190.92) and (335.14,190.96) .. (335,182.16) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (365.1,199.93) .. controls (374.95,191) and (374.81,190.96) .. (375,182.16) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (345.09,199.89) .. controls (355.05,206.7) and (355.05,206.7) .. (365.1,199.93) ;

(50.99,257.4) node [anchor=north west][inner sep=0.75pt]    $6)$;

(188.99,257.4) node [anchor=north west][inner sep=0.75pt]    $7)$;

(328.97,257.4) node [anchor=north west][inner sep=0.75pt]    $8)$;

(116,127.4) node [anchor=north west][inner sep=0.75pt]    $=$;

(256,127.4) node [anchor=north west][inner sep=0.75pt]    $+$;

So we obtain that diagram 6) is zero. Thus, diagram 1) is equal to diagram 4), or in other words, the path morphism of $P_2$ is equal to the path morphism of $P_1.$

$\bullet$ We have proved the proposition when $p_a=\mathbf{s}$ and $p_z$ is any vertex. One can prove similarly the proposition for $p_a=\mathbf{t}$ and $p_z$  any vertex, by symmetry. By flipping the diagrams we can prove the proposition for any $p_z\in \{\mathbf{s},\mathbf{t}\}$.

$\bullet$  The only case that remains to show is the equivalence for $p$ and $q$ such that $p_a=q_a=p_z=q_z=c$. There are four possible simplified paths

$$Q_1:=[c,\mathbf{s},c,\mathbf{t},c],  Q_2:=[c,\mathbf{t},c,\mathbf{s},c], Q_3:=[c,\mathbf{s},c,\mathbf{t},c,\mathbf{s},c], Q_4:=[c,\mathbf{t},c,\mathbf{s},c,\mathbf{t},c].$$

In the left hand-side of the following figure, we draw the path morphism corresponding to $Q_1$, which can be seen to be equal to the path morphism corresponding to $Q_2$ after an application of the local relation (2.15) in [1].

every picture/.style=line width=0.75pt

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (90,210) – (90,110.27) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (120,210) – (119.95,10) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (180,210) – (180.01,10) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (200,210) – (200.01,10) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (250,210) – (249.95,10) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (230,210) – (229.95,10) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (290,210) – (290,85) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (310,210) – (310,85) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (160,210) – (159.98,10) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (290.01,19.58) – (289.97,10) ;
[shift=(290.01,19.58), rotate = 269.73] [color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][fill=rgb, 255:red, 126; green, 211; blue, 33   ,fill opacity=1 ][line width=1.75]      (0, 0) circle [x radius= 4.62, y radius= 4.62]   ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (289.93,80.21) – (289.98,89.79) ;
[shift=(289.93,80.21), rotate = 89.73] [color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][fill=rgb, 255:red, 126; green, 211; blue, 33   ,fill opacity=1 ][line width=1.75]      (0, 0) circle [x radius= 4.62, y radius= 4.62]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (470,120) – (469.95,10) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (450,130) – (449.95,10) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (510.05,210) – (510.05,95) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (530,210) – (530,90) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (510.01,19.58) – (509.97,10) ;
[shift=(510.01,19.58), rotate = 269.73] [color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][fill=rgb, 255:red, 126; green, 211; blue, 33   ,fill opacity=1 ][line width=1.75]      (0, 0) circle [x radius= 4.62, y radius= 4.62]   ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (509.98,90.21) – (510.02,99.79) ;
[shift=(509.98,90.21), rotate = 89.73] [color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][fill=rgb, 255:red, 126; green, 211; blue, 33   ,fill opacity=1 ][line width=1.75]      (0, 0) circle [x radius= 4.62, y radius= 4.62]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (470,130) – (470,120) ;
[shift=(470,130), rotate = 270] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.75]      (0, 0) circle [x radius= 4.62, y radius= 4.62]   ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (270,210) – (270,85) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (140,210) – (140,10) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (360.05,130) – (360.05,120) ;
[shift=(360.05,130), rotate = 270] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.75]      (0, 0) circle [x radius= 4.62, y radius= 4.62]   ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (360.05,120) – (360,10) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (359.95,200.42) – (360,210) ;
[shift=(359.95,200.42), rotate = 89.73] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.75]      (0, 0) circle [x radius= 4.62, y radius= 4.62]   ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70,36.94) – (60,28.05) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70,60.27) – (70,36.94) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (60,28.05) .. controls (50.08,19.23) and (50.08,19.31) .. (50,10.27) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70,36.94) – (80,28.05) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (80,28.05) .. controls (89.92,19.23) and (89.92,19.31) .. (90,10.27) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (70,33.6) – (60,42.49) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (70,10.27) – (70,33.6) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (60,42.49) .. controls (50.08,51.31) and (50.08,51.23) .. (50,60.27) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (70,33.6) – (80,42.49) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (80,42.49) .. controls (89.92,51.31) and (89.92,51.23) .. (90,60.27) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (70,86.94) – (60,78.05) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (70,210) – (70,86.94) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (60,78.05) .. controls (50.08,69.23) and (50.08,69.31) .. (50,60.27) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (70,86.94) – (80,78.05) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (80,78.05) .. controls (89.92,69.23) and (89.92,69.31) .. (90,60.27) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70,83.6) – (60,92.49) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70,60.27) – (70,83.6) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (60,92.49) .. controls (50.08,101.31) and (50.08,101.23) .. (50,110.27) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70,83.6) – (80,92.49) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (80,92.49) .. controls (89.92,101.31) and (89.92,101.23) .. (90,110.27) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (30,136.67) – (20,127.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (30,160) – (30,136.67) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (20,127.78) .. controls (10.08,118.96) and (10.08,119.04) .. (10,110) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (30,136.67) – (40,127.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (40,127.78) .. controls (49.92,118.96) and (49.92,119.04) .. (50,110) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (30,133.33) – (20,142.22) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (30,110) – (30,133.33) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (20,142.22) .. controls (10.08,151.04) and (10.08,150.96) .. (10,160) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (30,133.33) – (40,142.22) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (40,142.22) .. controls (49.92,151.04) and (49.92,150.96) .. (50,160) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (30,186.67) – (20,177.78) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (30,210) – (30,186.67) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (20,177.78) .. controls (10.08,168.96) and (10.08,169.04) .. (10,160) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (30,186.67) – (40,177.78) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (40,177.78) .. controls (49.92,168.96) and (49.92,169.04) .. (50,160) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (30,183.33) – (20,192.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (30,160) – (30,183.33) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (20,192.22) .. controls (10.08,201.04) and (10.08,200.96) .. (10,210) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (30,183.33) – (40,192.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (40,192.22) .. controls (49.92,201.04) and (49.92,200.96) .. (50,210) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (10,110) – (10,10.27) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (30,110) – (30,10) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (290,63.33) – (280,72.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (290,50) – (290,63.33) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (280,72.22) .. controls (270.08,81.04) and (270.08,80.96) .. (270,90) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (290,63.33) – (300,72.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (300,72.22) .. controls (309.92,81.04) and (309.92,80.96) .. (310,90) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (290,36.67) – (280,27.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (290,50) – (290,36.67) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (280,27.78) .. controls (270.08,18.96) and (270.08,19.04) .. (270,10) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (290,36.67) – (300,27.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (300,27.78) .. controls (309.92,18.96) and (309.92,19.04) .. (310,10) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (380,10) – (380,135) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (340,10) – (340,135) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (360,156.67) – (350,147.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (360,170) – (360,156.67) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (350,147.78) .. controls (340.09,138.96) and (340.09,139.04) .. (340,130) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (360,156.67) – (370,147.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (370,147.78) .. controls (379.92,138.96) and (379.92,139.04) .. (380,130) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (360,183.33) – (350,192.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (360,170) – (360,183.33) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (350,192.22) .. controls (340.09,201.04) and (340.09,200.96) .. (340,210) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (360,183.33) – (370,192.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (370,192.22) .. controls (379.92,201.04) and (379.92,200.96) .. (380,210) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (399.99,210) – (400,10) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (419.99,210) – (420,10) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (469.95,200.42) – (470,210) ;
[shift=(469.95,200.42), rotate = 89.73] [color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][fill=rgb, 255:red, 6; green, 86; blue, 179   ,fill opacity=1 ][line width=1.75]      (0, 0) circle [x radius= 4.62, y radius= 4.62]   ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (470,156.67) – (460,147.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (470,170) – (470,156.67) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (460,147.78) .. controls (450.08,138.96) and (450.08,139.04) .. (450,130) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (470,156.67) – (480,147.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (480,147.78) .. controls (489.92,138.96) and (489.92,139.04) .. (490,130) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (470,183.33) – (460,192.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (470,170) – (470,183.33) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (460,192.22) .. controls (450.08,201.04) and (450.08,200.96) .. (450,210) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (470,183.33) – (480,192.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (480,192.22) .. controls (489.92,201.04) and (489.92,200.96) .. (490,210) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (510,36.67) – (500,27.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (510,50) – (510,36.67) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (500,27.78) .. controls (490.08,18.96) and (490.08,19.04) .. (490,10) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (510,36.67) – (520,27.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (520,27.78) .. controls (529.92,18.96) and (529.92,19.04) .. (530,10) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (510,63.33) – (500,72.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (510,50) – (510,63.33) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (500,72.22) .. controls (490.08,81.04) and (490.08,80.96) .. (490,90) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (510,63.33) – (520,72.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (520,72.22) .. controls (529.92,81.04) and (529.92,80.96) .. (530,90) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (490,130) – (490,90) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (650,10) – (650,109.73) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (630,183.06) – (620,191.95) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (630,159.73) – (630,183.06) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (620,191.95) .. controls (610.08,200.77) and (610.08,200.69) .. (610,209.73) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (630,183.06) – (640,191.95) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (640,191.95) .. controls (649.92,200.77) and (649.92,200.69) .. (650,209.73) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (630,186.4) – (620,177.51) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (630,209.73) – (630,186.4) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (620,177.51) .. controls (610.08,168.69) and (610.08,168.77) .. (610,159.73) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (630,186.4) – (640,177.51) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (640,177.51) .. controls (649.92,168.69) and (649.92,168.77) .. (650,159.73) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (630,133.06) – (620,141.95) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (630,10) – (630,133.06) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (620,141.95) .. controls (610.08,150.77) and (610.08,150.69) .. (610,159.73) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (630,133.06) – (640,141.95) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (640,141.95) .. controls (649.92,150.77) and (649.92,150.69) .. (650,159.73) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (630,136.4) – (620,127.51) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (630,159.73) – (630,136.4) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (620,127.51) .. controls (610.08,118.69) and (610.08,118.77) .. (610,109.73) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (630,136.4) – (640,127.51) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (640,127.51) .. controls (649.92,118.69) and (649.92,118.77) .. (650,109.73) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (590,83.33) – (580,92.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (590,60) – (590,83.33) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (580,92.22) .. controls (570.08,101.04) and (570.08,100.96) .. (570,110) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (590,83.33) – (600,92.22) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (600,92.22) .. controls (609.92,101.04) and (609.92,100.96) .. (610,110) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (590,86.67) – (580,77.78) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (590,110) – (590,86.67) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (580,77.78) .. controls (570.08,68.96) and (570.08,69.04) .. (570,60) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (590,86.67) – (600,77.78) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (600,77.78) .. controls (609.92,68.96) and (609.92,69.04) .. (610,60) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (590,33.33) – (580,42.22) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (590,10) – (590,33.33) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (580,42.22) .. controls (570.08,51.04) and (570.08,50.96) .. (570,60) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (590,33.33) – (600,42.22) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (600,42.22) .. controls (609.92,51.04) and (609.92,50.96) .. (610,60) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (590,36.67) – (580,27.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (590,60) – (590,36.67) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (580,27.78) .. controls (570.08,18.96) and (570.08,19.04) .. (570,10) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (590,36.67) – (600,27.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (600,27.78) .. controls (609.92,18.96) and (609.92,19.04) .. (610,10) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (570,110) – (570,209.73) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (590,110) – (590,210) ;

(95,100) node [anchor=north west][inner sep=0.75pt]   [align=left] =;

(205,100) node [anchor=north west][inner sep=0.75pt]   [align=left] +;

(425,100) node [anchor=north west][inner sep=0.75pt]   [align=left] +;

(315,100) node [anchor=north west][inner sep=0.75pt]   [align=left] +;

(535,110) node [anchor=north west][inner sep=0.75pt]   [align=left] =;

Proof that $f(Q_1)=f(Q_2).$

By reading Figure <ref> upside down, we have that $[\mathbf{s},c,\mathbf{t},c,\mathbf{s},c] \simeq [\mathbf{s},c,\mathbf{t},c]$. This way, we deduce that $Q_4 \simeq [c,\mathbf{t},c,\mathbf{s},c,\mathbf{t},c]
\simeq [c,\mathbf{t},c,\mathbf{s},c,\mathbf{t},c,\mathbf{s},c] \simeq [c,\mathbf{t},c,\mathbf{s},c] \simeq Q_2$. Analogously, $[c,\mathbf{s},c,\mathbf{t},c,\mathbf{s},c] \simeq [c,\mathbf{s},c,\mathbf{t},c]$, so $Q_3\simeq Q_1$.

The proof of the Forking Path Conjecture for the element $12132=12312$ is essentially the same as the proof given in Proposition <ref>, after applying the auto-equivalence of $\mathbb{S}Bim$ given by the unique non-trivial automorphism of the Dynkin diagram, i.e., applying a vertical symmetry to all diagrams.

§ THE COUNTEREXAMPLE

Let us consider the element $\sigma=12321 \in S_4$. The rex graph $Rex(\sigma)$ corresponds to the following figure:

every picture/.style=line width=0.75pt

[line width=1.5]
(160,40) – (190,10) ;

[line width=1.5]    (60,50) – (95,50) ;

[line width=1.5]
(260,10) – (290,40) ;

[line width=1.5]    (360,50) – (395,50) ;

[line width=1.5]
(260,90) – (290,60) ;

[line width=1.5]
(160,60) – (190,90) ;

(0,40) node [anchor=north west][inner sep=0.75pt]  [font=]  $12321$;

(195,0) node [anchor=north west][inner sep=0.75pt]  [font=]  $13213$;

(295,40) node [anchor=north west][inner sep=0.75pt]  [font=]  $31213$;

(100,40) node [anchor=north west][inner sep=0.75pt]  [font=]  $13231$;

(400,40) node [anchor=north west][inner sep=0.75pt]  [font=]  $32123$;

(195,85) node [anchor=north west][inner sep=0.75pt]  [font=]  $31231$;

Reduced expression graph of $\sigma$.

Note that the four vertices in the middle are the same vertex in the conflated expression graph.
Let us consider the element
$$x:=1 \otimes_{s_1} 1 \otimes_{s_3} 1 \otimes_{s_2}  x_3 \otimes_{s_3} 1 \otimes_{s_1} 1$$
in the Bott-Samelson bimodule $B_{1}B_{3}B_{2}B_{3}B_{1}$. Let $v_1$ and $v_2$ be the following paths respectively:

$v_1:=13231 \rightarrow  31231 \rightarrow  31213 \rightarrow  32123 \rightarrow  31213 \rightarrow  13213 \rightarrow  13231 \rightarrow  12321 \rightarrow  13231$.

$v_2:=13231 \rightarrow  12321 \rightarrow  13231 \rightarrow  31231 \rightarrow  31213 \rightarrow   32123 \rightarrow  31213 \rightarrow  13213 \rightarrow  13231$.

To simplify calculations we need the following. From Equation (<ref>) we obtain

\begin{equation}\label{eq3}
f_{i(i+1)}(1{\otimes_{s_i}}x_{i+1}{\otimes_{s_{i+1}}}1{\otimes_{s_i}}1) = 1{\otimes_{s_{i+1}}}1{\otimes_{s_i}}1{\otimes_{s_{i+1}}}x_{i+2}.
\end{equation}

Similarly, from Equation (<ref>) we obtain

\begin{equation}\label{eq6}
f_{i(i-1)}(1{\otimes_{s_i}}1{\otimes_{s_{i-1}}}x_{i}{\otimes_{s_i}}1) = x_{i-1}{\otimes_{s_{i-1}}}1{\otimes_{s_i}}1{\otimes_{s_{i-1}}}1.
\end{equation}

We will use a diagrammatic method to evaluate homomorphisms
in $\mathbb{S}Bim$. The next figure shows the evaluation of $f(v_1)$ (left) and $f(v_2)$ (right) in the element $x$ defined above.

every picture/.style=line width=0.75pt

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (115.92,373.83) .. controls (101.25,386.98) and (101,387.02) .. (100.79,399.93) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (25.13,373.9) .. controls (39.98,387.05) and (40.04,387.02) .. (40.26,399.93) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (70.53,74.62) – (70.53,152.7) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (55.39,61.61) – (70.53,74.62) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (85.66,87.64) .. controls (100.54,100.76) and (100.79,100.65) .. (100.79,113.66) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (55.39,87.64) .. controls (40.52,100.76) and (40.52,100.76) .. (40.26,113.66) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (85.66,61.61) – (70.53,74.62) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70.53,74.62) – (55.39,87.64) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70.53,74.62) – (85.66,87.64) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (55.39,61.61) .. controls (40.52,48.49) and (40.52,48.49) .. (40.26,35.59) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (85.66,61.61) .. controls (100.54,48.49) and (100.54,48.49) .. (100.79,35.59) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (10,9.56) – (10.21,165.62) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (131.05,9.56) – (131.12,165.83) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (10.25,217.79) – (10,347.88) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (130.98,217.88) – (131.05,347.88) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (115.92,191.82) – (70.63,152.7) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (55.53,139.66) .. controls (40.69,126.51) and (40.48,126.56) .. (40.26,113.66) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (85.79,139.71) .. controls (100.71,126.62) and (100.5,126.56) .. (100.79,113.66) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (25.13,191.64) – (70.63,152.7) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70.63,152.7) – (85.79,139.71) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70.63,152.7) – (55.53,139.66) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70.42,230.77) – (70.63,152.7) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (115.92,191.82) .. controls (130.76,204.97) and (130.76,204.97) .. (130.98,217.88) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (115.92,191.82) .. controls (130.83,178.73) and (130.83,178.73) .. (131.12,165.83) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (25.13,191.73) .. controls (10.47,204.88) and (10.47,204.88) .. (10.25,217.79) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (10.21,165.62) .. controls (10.08,178.83) and (9.97,178.74) .. (25.13,191.73) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (85.66,243.78) .. controls (100.54,256.9) and (100.79,256.79) .. (100.79,269.81) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (55.39,243.78) .. controls (40.52,256.9) and (40.52,256.9) .. (40.26,269.81) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70.53,230.77) – (55.39,243.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70.53,230.77) – (85.66,243.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (55.53,295.8) .. controls (40.69,282.65) and (40.48,282.71) .. (40.26,269.81) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (85.79,295.86) .. controls (100.71,282.77) and (100.5,282.71) .. (100.79,269.81) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70.63,308.84) – (85.79,295.86) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70.63,308.84) – (55.53,295.8) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70.53,308.84) – (70.53,399.93) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][line width=1.75]    (70.53,347.88) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (83.32,319.85) .. controls (98.33,333.04) and (98.33,333.04) .. (98.5,345.95) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (70.53,308.84) – (83.32,319.85) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (70.53,308.84) – (57.41,320.12) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (57.41,320.12) .. controls (42.53,333.24) and (42.53,333.24) .. (42.28,346.14) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (70.53,230.77) – (70.53,308.84) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (70.42,230.77) – (115.92,191.82) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (70.42,230.77) – (57.65,219.74) – (25.13,191.64) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (70.53,9.56) – (70.53,74.62) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (25.13,373.9) .. controls (10.25,360.78) and (10.25,360.78) .. (10,347.88) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (115.92,373.83) .. controls (130.8,360.71) and (130.8,360.71) .. (131.05,347.8) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (27.15,372.17) .. controls (12.27,385.29) and (10.25,387.02) .. (10,399.93) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (113.63,371.97) .. controls (128.64,385.17) and (130.88,387.02) .. (131.05,399.93) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (42.28,346.14) .. controls (42.43,359.35) and (42.47,359.28) .. (27.15,372.17) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (113.63,371.97) .. controls (108.35,367.43) and (104.91,364.47) .. (102.68,362) .. controls (98.5,357.37) and (98.5,354.42) .. (98.5,345.95) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (100.79,35.59) – (100.79,9.56) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (40.26,35.59) – (40.26,9.56) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (274.8,35.66) .. controls (260.13,22.51) and (259.88,22.47) .. (259.67,9.56) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (184.01,35.59) .. controls (198.86,22.44) and (198.92,22.47) .. (199.14,9.56) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (229.41,334.87) – (229.41,256.79) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (214.27,347.88) – (229.41,334.87) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (244.54,321.85) .. controls (259.42,308.73) and (259.67,308.84) .. (259.67,295.83) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (214.27,321.85) .. controls (199.4,308.73) and (199.4,308.73) .. (199.14,295.83) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (244.54,347.88) – (229.41,334.87) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (229.41,334.87) – (214.27,321.85) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (229.41,334.87) – (244.54,321.85) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (214.27,347.88) .. controls (199.4,361) and (199.4,361) .. (199.14,373.9) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (244.54,347.88) .. controls (259.42,361) and (259.42,361) .. (259.67,373.9) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (168.88,399.93) – (169.09,243.87) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (289.93,399.93) – (290,243.66) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (169.13,191.7) – (168.88,61.61) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (289.86,191.61) – (289.93,61.61) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (274.8,217.67) – (229.51,256.79) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (214.41,269.83) .. controls (199.57,282.98) and (199.36,282.93) .. (199.14,295.83) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (244.67,269.78) .. controls (259.59,282.87) and (259.38,282.93) .. (259.67,295.83) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (184.01,217.85) – (229.51,256.79) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (229.51,256.79) – (244.67,269.78) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (229.51,256.79) – (214.41,269.83) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (229.3,178.72) – (229.51,256.79) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (274.8,217.67) .. controls (289.64,204.52) and (289.64,204.52) .. (289.86,191.61) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (274.8,217.67) .. controls (289.71,230.76) and (289.71,230.76) .. (290,243.66) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (184.01,217.76) .. controls (169.35,204.61) and (169.35,204.61) .. (169.13,191.7) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (169.09,243.87) .. controls (168.96,230.66) and (168.85,230.75) .. (184.01,217.76) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (244.54,165.71) .. controls (259.42,152.59) and (259.67,152.7) .. (259.67,139.68) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (214.27,165.71) .. controls (199.4,152.59) and (199.4,152.59) .. (199.14,139.68) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (229.41,178.72) – (214.27,165.71) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (229.41,178.72) – (244.54,165.71) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (214.41,113.69) .. controls (199.57,126.84) and (199.36,126.78) .. (199.14,139.68) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (244.67,113.63) .. controls (259.59,126.72) and (259.38,126.78) .. (259.67,139.68) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (229.51,100.65) – (244.67,113.63) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (229.51,100.65) – (214.41,113.69) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (229.41,100.65) – (229.41,9.56) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][line width=1.75]    (229.41,61.61) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (242.2,89.64) .. controls (257.21,76.45) and (257.21,76.45) .. (257.38,63.54) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (229.41,100.65) – (242.2,89.64) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (229.41,100.65) – (216.29,89.37) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (216.29,89.37) .. controls (201.41,76.25) and (201.41,76.25) .. (201.16,63.35) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (229.41,178.72) – (229.41,100.65) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (229.3,178.72) – (274.8,217.67) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (229.3,178.72) – (216.53,189.75) – (184.01,217.85) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.75]    (229.41,399.93) – (229.41,334.87) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (184.01,35.59) .. controls (169.13,48.71) and (169.13,48.71) .. (168.88,61.61) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (274.8,35.66) .. controls (289.68,48.78) and (289.68,48.78) .. (289.93,61.69) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (186.03,37.32) .. controls (171.15,24.2) and (169.13,22.47) .. (168.88,9.56) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (272.51,37.52) .. controls (287.52,24.33) and (289.76,22.47) .. (289.93,9.56) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (201.16,63.35) .. controls (201.31,50.14) and (201.35,50.21) .. (186.03,37.32) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (272.51,37.52) .. controls (267.23,42.06) and (263.79,45.02) .. (261.56,47.49) .. controls (257.38,52.12) and (257.38,55.07) .. (257.38,63.54) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (259.67,373.9) – (259.67,399.93) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (199.14,373.9) – (199.14,399.93) ;

[dash pattern=on 0.84pt off 2.51pt]  (85.66,386.91) .. controls (79.04,353.21) and (125.69,357.33) .. (101.54,316.6) ;
[shift=(100.79,315.35), rotate = 58.43] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

[dash pattern=on 0.84pt off 2.51pt]  (100.79,302.34) .. controls (118.89,272.49) and (120.81,237.77) .. (95.77,196.26) ;
[shift=(95,195), rotate = 58.43] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

[dash pattern=on 0.84pt off 2.51pt]  (270,165) .. controls (288.85,139.67) and (278.17,72.51) .. (252.87,30.35) ;
[shift=(252.1,29.08), rotate = 58.43] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

[dash pattern=on 0.84pt off 2.51pt]  (75,185) .. controls (54.06,182.92) and (35.88,172.58) .. (35.02,156.98) ;
[shift=(35,155), rotate = 92.16] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

[dash pattern=on 0.84pt off 2.51pt]  (30,130) .. controls (21.75,87.21) and (53.36,80.4) .. (55.34,30.6) ;
[shift=(55.39,29.08), rotate = 91.55] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

[dash pattern=on 0.84pt off 2.51pt]  (221.84,204.75) .. controls (224.27,191.13) and (241.44,203.46) .. (254.04,186.37) ;
[shift=(255,185), rotate = 123.54] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

[dash pattern=on 0.84pt off 2.51pt]  (195,320) .. controls (167.41,280.5) and (207.79,276.26) .. (209.95,226.52) ;
[shift=(210,225), rotate = 91.55] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

[dash pattern=on 0.84pt off 2.51pt]  (244.54,386.91) .. controls (228.58,355.92) and (226.07,371.3) .. (205.93,341.4) ;
[shift=(205,340), rotate = 56.54] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

(25,135) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{2}$;

(85,300) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{3}$;

(260,167.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{3}$;

(75,385) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{3}$;

(77,178) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{3}$;

(200,209) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{2}$;

(234,7) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{3}$;

(181,322) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{2}$;

(45.16,8) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{2}$;

(234,385) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{3}$;

Diagrams for $v_1$ and $v_2$, evaluated in $x$.

We will prove that the elements obtained are different. It is known that

\begin{equation*}
B_3B_2B_3 \cong B_{323} \oplus B_3 \text{ , where } B_{323} \cong \langle1^{\otimes} \rangle \text{ and }  B_3 \cong \langle 1\otimes 1\otimes x_3 \otimes 1 \rangle .
\end{equation*}

So it is possible to generate the $R$-bimodule $B_3B_2B_3$ with the elements $1^{\otimes}$ and $1\otimes 1\otimes x_3 \otimes 1$. In $B_1B_3B_2B_3B_1$, we have

\begin{equation*}
f(v_1)(x) = 1 \otimes_{s_1} 1 \otimes_{s_3} 1 \otimes_{s_2} x_3 \otimes_{s_3} 1 \otimes_{s_1} 1
\end{equation*}

\begin{equation*}
f(v_2)(x) = 1 \otimes_{s_1} x_2 \otimes_{s_3} 1 \otimes_{s_2} 1 \otimes_{s_3} 1 \otimes_{s_1} 1.
\end{equation*}

If they were the same, applying dots over both $B_1$ in $B_1B_3B_2B_3B_1$ we would have that
$ x_2 \cdot 1^{\otimes} = 1\otimes_{s_3} 1 \otimes_{s_2} x_3 \otimes_{s_3} 1$ in $B_3B_2B_3$, which, as stated above, is not true.

§ A FAMILY OF COUNTEREXAMPLES

The element $\sigma=12321$ is the only one where the FPC fails for the group $S_4$. We proved this by showing that the diagrams in Figure <ref> (the same diagrams as in Figure <ref>) are not equal.

every picture/.style=line width=1pt

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (90,299.94) .. controls (80.31,310.05) and (80.14,310.08) .. (80,320) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (30,300) .. controls (39.81,310.11) and (39.86,310.08) .. (40,320) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (60,70) – (60,130) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (50,60) – (60,70) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (70,80) .. controls (79.83,90.08) and (80,90) .. (80,100) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (50,80) .. controls (40.17,90.08) and (40.17,90.08) .. (40,100) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (70,60) – (60,70) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (60,70) – (50,80) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (60,70) – (70,80) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (50,60) .. controls (40.17,49.92) and (40.17,49.92) .. (40,40) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (70,60) .. controls (79.83,49.92) and (79.83,49.92) .. (80,40) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (20,20) – (20.14,139.93) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (100,20) – (100.05,140.09) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (20.17,180.02) – (20,280) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (99.95,180.09) – (100,280) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (90,160.07) – (60.07,130) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (50.09,119.98) .. controls (40.28,109.87) and (40.14,109.92) .. (40,100) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (70.09,120.02) .. controls (79.95,109.96) and (79.81,109.92) .. (80,100) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (30,159.93) – (60.07,130) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (60.07,130) – (70.09,120.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (60.07,130) – (50.09,119.98) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (59.93,190) – (60.07,130) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (90,160.07) .. controls (99.81,170.17) and (99.81,170.17) .. (99.95,180.09) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (90,160.07) .. controls (99.86,150.01) and (99.86,150.01) .. (100.05,140.09) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (30,160) .. controls (20.31,170.11) and (20.31,170.11) .. (20.17,180.02) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (20.14,139.93) .. controls (20.06,150.08) and (19.98,150.02) .. (30,160) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (70,200) .. controls (79.83,210.08) and (80,210) .. (80,220) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (50,200) .. controls (40.17,210.08) and (40.17,210.08) .. (40,220) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (60,190) – (50,200) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (60,190) – (70,200) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (50.09,239.98) .. controls (40.28,229.87) and (40.14,229.92) .. (40,220) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (70.09,240.02) .. controls (79.95,229.96) and (79.81,229.92) .. (80,220) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (60.07,250) – (70.09,240.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (60.07,250) – (50.09,239.98) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (60,250) – (60,320) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][line width=1.5]    (60,280) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (68.46,258.46) .. controls (78.38,268.6) and (78.38,268.6) .. (78.49,278.52) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (60,250) – (68.46,258.46) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (60,250) – (51.33,258.67) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (51.33,258.67) .. controls (41.5,268.75) and (41.5,268.75) .. (41.33,278.67) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (60,190) – (60,250) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (59.93,190) – (90,160.07) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (59.93,190) – (51.49,181.52) – (30,159.93) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (60,20) – (60,70) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (30,300) .. controls (20.17,289.92) and (20.17,289.92) .. (20,280) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (90,299.94) .. controls (99.83,289.86) and (99.83,289.86) .. (100,279.94) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (31.33,298.67) .. controls (21.5,308.75) and (20.17,310.08) .. (20,320) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (88.49,298.52) .. controls (98.41,308.66) and (99.89,310.08) .. (100,320) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (41.33,278.67) .. controls (41.43,288.82) and (41.46,288.76) .. (31.33,298.67) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (88.49,298.52) .. controls (85,295.03) and (82.73,292.75) .. (81.25,290.85) .. controls (78.49,287.3) and (78.49,285.03) .. (78.49,278.52) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (80,40) – (80,20) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (40,40) – (40,20) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (240,40.06) .. controls (230.31,29.95) and (230.14,29.92) .. (230,20) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (180,40) .. controls (189.81,29.89) and (189.86,29.92) .. (190,20) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (210,270) – (210,210) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (200,280) – (210,270) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (220,260) .. controls (229.83,249.92) and (230,250) .. (230,240) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (200,260) .. controls (190.17,249.92) and (190.17,249.92) .. (190,240) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (220,280) – (210,270) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (210,270) – (200,260) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (210,270) – (220,260) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (200,280) .. controls (190.17,290.08) and (190.17,290.08) .. (190,300) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (220,280) .. controls (229.83,290.08) and (229.83,290.08) .. (230,300) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (170,320) – (170.14,200.07) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (250,320) – (250.05,199.91) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (170.17,159.98) – (170,60) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (249.95,159.91) – (250,60) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (240,179.93) – (210.07,210) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (200.09,220.02) .. controls (190.28,230.13) and (190.14,230.08) .. (190,240) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (220.09,219.98) .. controls (229.95,230.04) and (229.81,230.08) .. (230,240) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (180,180.07) – (210.07,210) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (210.07,210) – (220.09,219.98) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (210.07,210) – (200.09,220.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (209.93,150) – (210.07,210) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (240,179.93) .. controls (249.81,169.83) and (249.81,169.83) .. (249.95,159.91) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (240,179.93) .. controls (249.86,189.99) and (249.86,189.99) .. (250.05,199.91) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (180,180) .. controls (170.31,169.89) and (170.31,169.89) .. (170.17,159.98) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (170.14,200.07) .. controls (170.06,189.92) and (169.98,189.98) .. (180,180) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (220,140) .. controls (229.83,129.92) and (230,130) .. (230,120) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (200,140) .. controls (190.17,129.92) and (190.17,129.92) .. (190,120) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (210,150) – (200,140) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (210,150) – (220,140) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (200.09,100.02) .. controls (190.28,110.13) and (190.14,110.08) .. (190,120) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (220.09,99.98) .. controls (229.95,110.04) and (229.81,110.08) .. (230,120) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (210.07,90) – (220.09,99.98) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (210.07,90) – (200.09,100.02) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (210,90) – (210,20) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][line width=1.5]    (210,60) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (218.46,81.54) .. controls (228.38,71.4) and (228.38,71.4) .. (228.49,61.48) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (210,90) – (218.46,81.54) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (210,90) – (201.33,81.33) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (201.33,81.33) .. controls (191.5,71.25) and (191.5,71.25) .. (191.33,61.33) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (210,150) – (210,90) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (209.93,150) – (240,179.93) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (209.93,150) – (201.49,158.48) – (180,180.07) ;

[color=rgb, 255:red, 255; green, 0; blue, 0   ,draw opacity=1 ][line width=1.5]    (210,320) – (210,270) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (180,40) .. controls (170.17,50.08) and (170.17,50.08) .. (170,60) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (240,40.06) .. controls (249.83,50.14) and (249.83,50.14) .. (250,60.06) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (181.33,41.33) .. controls (171.5,31.25) and (170.17,29.92) .. (170,20) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (238.49,41.48) .. controls (248.41,31.34) and (249.89,29.92) .. (250,20) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (191.33,61.33) .. controls (191.43,51.18) and (191.46,51.24) .. (181.33,41.33) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.5]    (238.49,41.48) .. controls (235,44.97) and (232.73,47.25) .. (231.25,49.15) .. controls (228.49,52.7) and (228.49,54.97) .. (228.49,61.48) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (230,300) – (230,320) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.5]    (190,300) – (190,320) ;

(110,160) node [anchor=north west][inner sep=0.75pt]    and;

Diagrams for $f(v_1)$ and $f(v_2)$.

The first diagram decomposes as in Figure <ref>, while the second decomposes as in Figure <ref>. The only summand that is different is the last one.

every picture/.style=line width=0.75pt

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (89.91,40) .. controls (80.22,50.11) and (80.22,50.11) .. (80.07,60.02) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (89.95,160) .. controls (80.12,149.92) and (80.12,149.92) .. (79.95,140) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (29.95,160) .. controls (39.79,149.92) and (39.79,149.92) .. (39.95,140) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (30,39.98) .. controls (39.81,50.08) and (39.81,50.08) .. (39.95,60) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (29.95,40) .. controls (20.12,29.92) and (20.12,29.92) .. (19.95,20) ;

[color=rgb, 255:red, 6; green, 86; blue, 179   ,draw opacity=1 ][line width=1.75]    (89.95,40) .. controls (99.79,29.92) and (99.79,29.92) .. (99.95,20) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (29.95,40) .. controls (20.12,50.08) and (20.12,50.08) .. (19.95,60) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (89.95,40) .. controls (99.87,50.14) and (99.87,50.14) .. (99.98,60.06) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (19.95,60) – (20.09,139.93) ;

[color=rgb, 255:red, 126; green, 211; blue, 33   ,draw opacity=1 ][line width=1.75]    (99.98,60.06) – (100,140.09) ;