CONFIGURATION: own coordinates (verifier C)
  punctures: {'a': ('0', '4'), 'b': ('-4', '-3'), 'c': ('4', '-3'), 'tI': ('0', '0'), 'tO': ('0', '-8')}
  arc alpha_1: [('4', '-3'), ('0', '4')]  radius about 0.1
  arc alpha_2: [('0', '0'), ('0', '-8')]  radius about 0.13
  arc alpha_3: [('0', '4'), ('-4', '-3')]  radius about 0.16
  arc alpha_4: [('0', '0'), ('6', '3'), ('8', '-3'), ('4', '-9'), ('0', '-8')]  radius about 0.19
  arc alpha_5: [('-4', '-3'), ('4', '-3')]  radius about 0.22
  arc alpha_6: [('0', '0'), ('-6', '3'), ('-17/2', '-5/2'), ('-9/2', '-19/2'), ('0', '-8')]  radius about 0.25

(0) the picture described in Section 8.3
  [ok] t_I inside, t_O outside the triangle abc
  [ok] alpha_2 and the side alpha_1: 0 crossing(s), no common end point
  [ok] alpha_2 and the side alpha_3: 0 crossing(s), no common end point
  [ok] alpha_2 and the side alpha_5: 1 crossing(s), no common end point
  [ok] alpha_4 and the side alpha_1: 1 crossing(s), no common end point
  [ok] alpha_4 and the side alpha_3: 0 crossing(s), no common end point
  [ok] alpha_4 and the side alpha_5: 0 crossing(s), no common end point
  [ok] alpha_6 and the side alpha_1: 0 crossing(s), no common end point
  [ok] alpha_6 and the side alpha_3: 1 crossing(s), no common end point
  [ok] alpha_6 and the side alpha_5: 0 crossing(s), no common end point
  [ok] alpha_2, alpha_4: disjoint except for t_I, t_O
  [ok] alpha_4, alpha_6: disjoint except for t_I, t_O
  [ok] alpha_2, alpha_6: disjoint except for t_I, t_O
  [ok] sides alpha_1, alpha_3: one common vertex
  [ok] sides alpha_3, alpha_5: one common vertex
  [ok] sides alpha_1, alpha_5: one common vertex
  [ok] D_1 is a simple polygon (a closed disc) and D_1 n P = ['a', 'c']; alpha_1 lies in its interior
  [ok] D_2 is a simple polygon (a closed disc) and D_2 n P = ['tI', 'tO']; alpha_2 lies in its interior
  [ok] D_3 is a simple polygon (a closed disc) and D_3 n P = ['a', 'b']; alpha_3 lies in its interior
  [ok] D_4 is a simple polygon (a closed disc) and D_4 n P = ['tI', 'tO']; alpha_4 lies in its interior
  [ok] D_5 is a simple polygon (a closed disc) and D_5 n P = ['b', 'c']; alpha_5 lies in its interior
  [ok] D_6 is a simple polygon (a closed disc) and D_6 n P = ['tI', 'tO']; alpha_6 lies in its interior

(H1)/(H2): the fifteen pairs
  [ok] (H1) D_1 n D_2 is empty (hexagon edge 1-2)
  [ok] (H2) bd D_1, bd D_3: 2 transverse crossings, 4 components: 2 corners, punctures {a}; 2 corners, punctures {b}; 2 corners, punctures {c}; 2 corners, punctures {tI,tO} (outer) -- no component with two corners and no puncture
  [ok] (H2) bd D_1, bd D_4: 4 transverse crossings, 6 components: 2 corners, punctures {a}; 2 corners, punctures {c}; 2 corners, punctures {tI}; 2 corners, punctures {tO}; 4 corners, punctures {}; 4 corners, punctures {b} (outer) -- no component with two corners and no puncture
  [ok] (H2) bd D_1, bd D_5: 2 transverse crossings, 4 components: 2 corners, punctures {a}; 2 corners, punctures {b}; 2 corners, punctures {c}; 2 corners, punctures {tI,tO} (outer) -- no component with two corners and no puncture
  [ok] (H1) D_1 n D_6 is empty (hexagon edge 1-6)
  [ok] (H1) D_2 n D_3 is empty (hexagon edge 2-3)
  [ok] (H2) bd D_2, bd D_4: 4 transverse crossings, 6 components: 2 corners, punctures {a,b} (outer); 2 corners, punctures {c}; 2 corners, punctures {tI}; 2 corners, punctures {tO}; 4 corners, punctures {}; 4 corners, punctures {} -- no component with two corners and no puncture
  [ok] (H2) bd D_2, bd D_5: 4 transverse crossings, 6 components: 2 corners, punctures {b}; 2 corners, punctures {c}; 2 corners, punctures {tI}; 2 corners, punctures {tO}; 4 corners, punctures {}; 4 corners, punctures {a} (outer) -- no component with two corners and no puncture
  [ok] (H2) bd D_2, bd D_6: 4 transverse crossings, 6 components: 2 corners, punctures {a,c} (outer); 2 corners, punctures {b}; 2 corners, punctures {tI}; 2 corners, punctures {tO}; 4 corners, punctures {}; 4 corners, punctures {} -- no component with two corners and no puncture
  [ok] (H1) D_3 n D_4 is empty (hexagon edge 3-4)
  [ok] (H2) bd D_3, bd D_5: 2 transverse crossings, 4 components: 2 corners, punctures {a}; 2 corners, punctures {b}; 2 corners, punctures {c}; 2 corners, punctures {tI,tO} (outer) -- no component with two corners and no puncture
  [ok] (H2) bd D_3, bd D_6: 4 transverse crossings, 6 components: 2 corners, punctures {a}; 2 corners, punctures {b}; 2 corners, punctures {tI}; 2 corners, punctures {tO}; 4 corners, punctures {}; 4 corners, punctures {c} (outer) -- no component with two corners and no puncture
  [ok] (H1) D_4 n D_5 is empty (hexagon edge 4-5)
  [ok] (H2) bd D_4, bd D_6: 4 transverse crossings, 6 components: 2 corners, punctures {a} (outer); 2 corners, punctures {b,c}; 2 corners, punctures {tI}; 2 corners, punctures {tO}; 4 corners, punctures {}; 4 corners, punctures {} -- no component with two corners and no puncture
  [ok] (H1) D_5 n D_6 is empty (hexagon edge 5-6)
  crossings per pair: {(1, 3): 2, (1, 4): 4, (1, 5): 2, (2, 4): 4, (2, 5): 4, (2, 6): 4, (3, 5): 2, (3, 6): 4, (4, 6): 4}

(H3): the five minimal sets, and all 63 non-empty sets Y
  [ok] (H3) Y=(1, 4): 4 crossings, 6 components (= crossings + 2), union of curves connected, punctures per component: [0, 1, 1, 1, 1, 1]
  [ok] (H3) Y=(2, 5): 4 crossings, 6 components (= crossings + 2), union of curves connected, punctures per component: [0, 1, 1, 1, 1, 1]
  [ok] (H3) Y=(3, 6): 4 crossings, 6 components (= crossings + 2), union of curves connected, punctures per component: [0, 1, 1, 1, 1, 1]
  [ok] (H3) Y=(1, 3, 5): 6 crossings, 8 components (= crossings + 2), union of curves connected, punctures per component: [0, 0, 0, 1, 1, 1, 1, 1]
  [ok] (H3) Y=(2, 4, 6): 12 crossings, 14 components (= crossings + 2), union of curves connected, punctures per component: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]
  [ok] Lemma 3.1 for C_6: 'not contained in a join' = 'connected and dominating in the prism' for all 63 sets
  [ok] number of non-empty vertex sets not contained in a join of C_6: 39
  [ok] every set not contained in a join contains one of the five sets, and these five are not contained in a join
  [ok] the hypothesis of Lemma 6.3(4) (all complementary components are discs with <= 1 puncture) holds for exactly the 39 sets not contained in a join, and fails for the 24 sets contained in a join

full arrangement of the six curves
  [ok] 30 crossings, 32 faces, all discs with at most one puncture
  the face of each puncture lies in the discs: {'c': [1, 5], 'a': [1, 3], 'tI': [2, 4, 6], 'tO': [2, 4, 6], 'b': [3, 5]}
  faces inside D_1 n D_3 n D_5: 0 ;  inside D_2 n D_4 n D_6: 2  (Remark 8.8: not pointed)

cyclic covers (Theorem 8.7)
  [ok] d= 5, weights (a,b,c,tI,tO)=(1,1,1,1,1): genus 6 = 3(d-1)/2; X_j (genus, boundary curves) = {1: (2, 1), 2: (2, 1), 3: (2, 1), 4: (2, 1), 5: (2, 1), 6: (2, 1)}; chi(X_j)=2-d; lifted bigons for the 9 pairs: 0; lifted curves of Y cut the cover into discs exactly for the 39 sets: True
  [ok] d= 7, weights (a,b,c,tI,tO)=(1,1,1,1,3): genus 9 = 3(d-1)/2; X_j (genus, boundary curves) = {1: (3, 1), 2: (3, 1), 3: (3, 1), 4: (3, 1), 5: (3, 1), 6: (3, 1)}; chi(X_j)=2-d; lifted bigons for the 9 pairs: 0; lifted curves of Y cut the cover into discs exactly for the 39 sets: True
  [ok] d= 9, weights (a,b,c,tI,tO)=(1,1,1,1,5): genus 12 = 3(d-1)/2; X_j (genus, boundary curves) = {1: (4, 1), 2: (3, 3), 3: (4, 1), 4: (3, 3), 5: (4, 1), 6: (3, 3)}; chi(X_j)=2-d; lifted bigons for the 9 pairs: 0; lifted curves of Y cut the cover into discs exactly for the 39 sets: True
  [ok] d=11, weights (a,b,c,tI,tO)=(1,1,1,1,7): genus 15 = 3(d-1)/2; X_j (genus, boundary curves) = {1: (5, 1), 2: (5, 1), 3: (5, 1), 4: (5, 1), 5: (5, 1), 6: (5, 1)}; chi(X_j)=2-d; lifted bigons for the 9 pairs: 0; lifted curves of Y cut the cover into discs exactly for the 39 sets: True
  [ok] d=13, weights (a,b,c,tI,tO)=(1,1,1,1,9): genus 18 = 3(d-1)/2; X_j (genus, boundary curves) = {1: (6, 1), 2: (6, 1), 3: (6, 1), 4: (6, 1), 5: (6, 1), 6: (6, 1)}; chi(X_j)=2-d; lifted bigons for the 9 pairs: 0; lifted curves of Y cut the cover into discs exactly for the 39 sets: True
  [ok] d=15, weights (a,b,c,tI,tO)=(1,1,1,1,11): genus 21 = 3(d-1)/2; X_j (genus, boundary curves) = {1: (7, 1), 2: (6, 3), 3: (7, 1), 4: (6, 3), 5: (7, 1), 6: (6, 3)}; chi(X_j)=2-d; lifted bigons for the 9 pairs: 0; lifted curves of Y cut the cover into discs exactly for the 39 sets: True
  [ok] d=21, weights (a,b,c,tI,tO)=(1,1,1,1,17): genus 30 = 3(d-1)/2; X_j (genus, boundary curves) = {1: (10, 1), 2: (9, 3), 3: (10, 1), 4: (9, 3), 5: (10, 1), 6: (9, 3)}; chi(X_j)=2-d; lifted bigons for the 9 pairs: 0; lifted curves of Y cut the cover into discs exactly for the 39 sets: True
  [ok] d=25, weights (a,b,c,tI,tO)=(1,1,1,1,21): genus 36 = 3(d-1)/2; X_j (genus, boundary curves) = {1: (12, 1), 2: (12, 1), 3: (12, 1), 4: (12, 1), 5: (12, 1), 6: (12, 1)}; chi(X_j)=2-d; lifted bigons for the 9 pairs: 0; lifted curves of Y cut the cover into discs exactly for the 39 sets: True

RESULT: ALL CHECKS PASSED
