pentagon picture: punctures inside each disc: {1: [('s', 1), ('s', 5)], 2: [('s', 1), ('s', 2)], 3: [('s', 2), ('s', 3)], 4: [('s', 3), ('s', 4)], 5: [('s', 4), ('s', 5)]}
====================================================================================================
PENTAGON PICTURE (Section 8, m = 5): Gamma = complement of C_5 = pentagon
degree d = 5, exponents w = {('s', 1): 1, ('s', 2): 1, ('s', 3): 1, ('s', 4): 1, ('s', 5): 1}
cover: chi = -10, genus = 6 ; formula (d-1)(|P|-2)/2 = 6
  X_1 = lift of the disc around [('s', 1), ('s', 5)]: connected True, chi -3, boundary curves 1 (gcd formula 1), genus 2, xi 4 ; complement chi [-7]
  X_2 = lift of the disc around [('s', 1), ('s', 2)]: connected True, chi -3, boundary curves 1 (gcd formula 1), genus 2, xi 4 ; complement chi [-7]
  X_3 = lift of the disc around [('s', 2), ('s', 3)]: connected True, chi -3, boundary curves 1 (gcd formula 1), genus 2, xi 4 ; complement chi [-7]
  X_4 = lift of the disc around [('s', 3), ('s', 4)]: connected True, chi -3, boundary curves 1 (gcd formula 1), genus 2, xi 4 ; complement chi [-7]
  X_5 = lift of the disc around [('s', 4), ('s', 5)]: connected True, chi -3, boundary curves 1 (gcd formula 1), genus 2, xi 4 ; complement chi [-7]
  pair (1,2) must OVERLAP : regions (chi, corners) [(-3, 10), (1, 10), (1, 10), (1, 10)] -> ok
  pair (1,3) must be DISJOINT: regions (chi, corners) [(-4, 0), (-3, 0), (-3, 0)] -> ok
  pair (1,4) must be DISJOINT: regions (chi, corners) [(-4, 0), (-3, 0), (-3, 0)] -> ok
  pair (1,5) must OVERLAP : regions (chi, corners) [(-3, 10), (1, 10), (1, 10), (1, 10)] -> ok
  pair (2,3) must OVERLAP : regions (chi, corners) [(-3, 10), (1, 10), (1, 10), (1, 10)] -> ok
  pair (2,4) must be DISJOINT: regions (chi, corners) [(-4, 0), (-3, 0), (-3, 0)] -> ok
  pair (2,5) must be DISJOINT: regions (chi, corners) [(-4, 0), (-3, 0), (-3, 0)] -> ok
  pair (3,4) must OVERLAP : regions (chi, corners) [(-3, 10), (1, 10), (1, 10), (1, 10)] -> ok
  pair (3,5) must be DISJOINT: regions (chi, corners) [(-4, 0), (-3, 0), (-3, 0)] -> ok
  pair (4,5) must OVERLAP : regions (chi, corners) [(-3, 10), (1, 10), (1, 10), (1, 10)] -> ok
  Y = [3, 4, 5] (not in a join): planar regions (chi, punctures) [(1, ()), (1, (('s', 1),)), (1, (('s', 2),)), (1, (('s', 3),)), (1, (('s', 4),)), (1, (('s', 5),))] ; lifted regions 10, all discs: True
  Y = [4, 5, 1] (not in a join): planar regions (chi, punctures) [(1, ()), (1, (('s', 1),)), (1, (('s', 2),)), (1, (('s', 3),)), (1, (('s', 4),)), (1, (('s', 5),))] ; lifted regions 10, all discs: True
  Y = [5, 1, 2] (not in a join): planar regions (chi, punctures) [(1, ()), (1, (('s', 1),)), (1, (('s', 2),)), (1, (('s', 3),)), (1, (('s', 4),)), (1, (('s', 5),))] ; lifted regions 10, all discs: True
  Y = [1, 2, 3] (not in a join): planar regions (chi, punctures) [(1, ()), (1, (('s', 1),)), (1, (('s', 2),)), (1, (('s', 3),)), (1, (('s', 4),)), (1, (('s', 5),))] ; lifted regions 10, all discs: True
  Y = [2, 3, 4] (not in a join): planar regions (chi, punctures) [(1, ()), (1, (('s', 1),)), (1, (('s', 2),)), (1, (('s', 3),)), (1, (('s', 4),)), (1, (('s', 5),))] ; lifted regions 10, all discs: True
  control Y = [4, 5] (inside a join): lifted regions chi [-3, 1] -> not all discs: True
  control Y = [5, 1] (inside a join): lifted regions chi [-3, 1] -> not all discs: True
  control Y = [1, 2] (inside a join): lifted regions chi [-3, 1] -> not all discs: True
  control Y = [2, 3] (inside a join): lifted regions chi [-3, 1] -> not all discs: True
  control Y = [3, 4] (inside a join): lifted regions chi [-3, 1] -> not all discs: True
RESULT for this configuration: ALL CHECKS PASSED
====================================================================================================
PENTAGON PICTURE (Section 8, m = 5): Gamma = complement of C_5 = pentagon
degree d = 7, exponents w = {('s', 1): 1, ('s', 2): 1, ('s', 3): 1, ('s', 4): 1, ('s', 5): 3}
cover: chi = -16, genus = 9 ; formula (d-1)(|P|-2)/2 = 9
  X_1 = lift of the disc around [('s', 1), ('s', 5)]: connected True, chi -5, boundary curves 1 (gcd formula 1), genus 3, xi 7 ; complement chi [-11]
  X_2 = lift of the disc around [('s', 1), ('s', 2)]: connected True, chi -5, boundary curves 1 (gcd formula 1), genus 3, xi 7 ; complement chi [-11]
  X_3 = lift of the disc around [('s', 2), ('s', 3)]: connected True, chi -5, boundary curves 1 (gcd formula 1), genus 3, xi 7 ; complement chi [-11]
  X_4 = lift of the disc around [('s', 3), ('s', 4)]: connected True, chi -5, boundary curves 1 (gcd formula 1), genus 3, xi 7 ; complement chi [-11]
  X_5 = lift of the disc around [('s', 4), ('s', 5)]: connected True, chi -5, boundary curves 1 (gcd formula 1), genus 3, xi 7 ; complement chi [-11]
  pair (1,2) must OVERLAP : regions (chi, corners) [(-5, 14), (1, 14), (1, 14), (1, 14)] -> ok
  pair (1,3) must be DISJOINT: regions (chi, corners) [(-6, 0), (-5, 0), (-5, 0)] -> ok
  pair (1,4) must be DISJOINT: regions (chi, corners) [(-6, 0), (-5, 0), (-5, 0)] -> ok
  pair (1,5) must OVERLAP : regions (chi, corners) [(-5, 14), (1, 14), (1, 14), (1, 14)] -> ok
  pair (2,3) must OVERLAP : regions (chi, corners) [(-5, 14), (1, 14), (1, 14), (1, 14)] -> ok
  pair (2,4) must be DISJOINT: regions (chi, corners) [(-6, 0), (-5, 0), (-5, 0)] -> ok
  pair (2,5) must be DISJOINT: regions (chi, corners) [(-6, 0), (-5, 0), (-5, 0)] -> ok
  pair (3,4) must OVERLAP : regions (chi, corners) [(-5, 14), (1, 14), (1, 14), (1, 14)] -> ok
  pair (3,5) must be DISJOINT: regions (chi, corners) [(-6, 0), (-5, 0), (-5, 0)] -> ok
  pair (4,5) must OVERLAP : regions (chi, corners) [(-5, 14), (1, 14), (1, 14), (1, 14)] -> ok
  Y = [3, 4, 5] (not in a join): planar regions (chi, punctures) [(1, ()), (1, (('s', 1),)), (1, (('s', 2),)), (1, (('s', 3),)), (1, (('s', 4),)), (1, (('s', 5),))] ; lifted regions 12, all discs: True
  Y = [4, 5, 1] (not in a join): planar regions (chi, punctures) [(1, ()), (1, (('s', 1),)), (1, (('s', 2),)), (1, (('s', 3),)), (1, (('s', 4),)), (1, (('s', 5),))] ; lifted regions 12, all discs: True
  Y = [5, 1, 2] (not in a join): planar regions (chi, punctures) [(1, ()), (1, (('s', 1),)), (1, (('s', 2),)), (1, (('s', 3),)), (1, (('s', 4),)), (1, (('s', 5),))] ; lifted regions 12, all discs: True
  Y = [1, 2, 3] (not in a join): planar regions (chi, punctures) [(1, ()), (1, (('s', 1),)), (1, (('s', 2),)), (1, (('s', 3),)), (1, (('s', 4),)), (1, (('s', 5),))] ; lifted regions 12, all discs: True
  Y = [2, 3, 4] (not in a join): planar regions (chi, punctures) [(1, ()), (1, (('s', 1),)), (1, (('s', 2),)), (1, (('s', 3),)), (1, (('s', 4),)), (1, (('s', 5),))] ; lifted regions 12, all discs: True
  control Y = [4, 5] (inside a join): lifted regions chi [-5, 1] -> not all discs: True
  control Y = [5, 1] (inside a join): lifted regions chi [-5, 1] -> not all discs: True
  control Y = [1, 2] (inside a join): lifted regions chi [-5, 1] -> not all discs: True
  control Y = [2, 3] (inside a join): lifted regions chi [-5, 1] -> not all discs: True
  control Y = [3, 4] (inside a join): lifted regions chi [-5, 1] -> not all discs: True
RESULT for this configuration: ALL CHECKS PASSED

hexagon picture: punctures inside each disc: {1: ['a', 'c'], 3: ['a', 'b'], 5: ['b', 'c'], 2: ['tI', 'tO'], 4: ['tI', 'tO'], 6: ['tI', 'tO']}
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HEXAGON PICTURE (triangle a,b,c and three spokes tI -> tO): Gamma = C_6
degree d = 5, exponents w = {'a': 1, 'b': 1, 'c': 1, 'tI': 1, 'tO': 1}
cover: chi = -10, genus = 6 ; formula (d-1)(|P|-2)/2 = 6
  X_1 = lift of the disc around ['a', 'c']: connected True, chi -3, boundary curves 1 (gcd formula 1), genus 2, xi 4 ; complement chi [-7]
  X_2 = lift of the disc around ['tI', 'tO']: connected True, chi -3, boundary curves 1 (gcd formula 1), genus 2, xi 4 ; complement chi [-7]
  X_3 = lift of the disc around ['a', 'b']: connected True, chi -3, boundary curves 1 (gcd formula 1), genus 2, xi 4 ; complement chi [-7]
  X_4 = lift of the disc around ['tI', 'tO']: connected True, chi -3, boundary curves 1 (gcd formula 1), genus 2, xi 4 ; complement chi [-7]
  X_5 = lift of the disc around ['b', 'c']: connected True, chi -3, boundary curves 1 (gcd formula 1), genus 2, xi 4 ; complement chi [-7]
  X_6 = lift of the disc around ['tI', 'tO']: connected True, chi -3, boundary curves 1 (gcd formula 1), genus 2, xi 4 ; complement chi [-7]
  pair (1,2) must be DISJOINT: regions (chi, corners) [(-4, 0), (-3, 0), (-3, 0)] -> ok
  pair (1,3) must OVERLAP : regions (chi, corners) [(-3, 10), (1, 10), (1, 10), (1, 10)] -> ok
  pair (1,4) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 10), (1, 10), (1, 10), (1, 10), (1, 20)] -> ok
  pair (1,5) must OVERLAP : regions (chi, corners) [(-3, 10), (1, 10), (1, 10), (1, 10)] -> ok
  pair (1,6) must be DISJOINT: regions (chi, corners) [(-4, 0), (-3, 0), (-3, 0)] -> ok
  pair (2,3) must be DISJOINT: regions (chi, corners) [(-4, 0), (-3, 0), (-3, 0)] -> ok
  pair (2,4) must OVERLAP : regions (chi, corners) [(-3, 10), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 10), (1, 10), (1, 10)] -> ok
  pair (2,5) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 10), (1, 10), (1, 10), (1, 10), (1, 20)] -> ok
  pair (2,6) must OVERLAP : regions (chi, corners) [(-3, 10), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 10), (1, 10), (1, 10)] -> ok
  pair (3,4) must be DISJOINT: regions (chi, corners) [(-4, 0), (-3, 0), (-3, 0)] -> ok
  pair (3,5) must OVERLAP : regions (chi, corners) [(-3, 10), (1, 10), (1, 10), (1, 10)] -> ok
  pair (3,6) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 10), (1, 10), (1, 10), (1, 10), (1, 20)] -> ok
  pair (4,5) must be DISJOINT: regions (chi, corners) [(-4, 0), (-3, 0), (-3, 0)] -> ok
  pair (4,6) must OVERLAP : regions (chi, corners) [(-3, 10), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 10), (1, 10), (1, 10)] -> ok
  pair (5,6) must be DISJOINT: regions (chi, corners) [(-4, 0), (-3, 0), (-3, 0)] -> ok
  Y = [1, 4] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 10, all discs: True
  Y = [2, 5] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 10, all discs: True
  Y = [3, 6] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 10, all discs: True
  Y = [1, 3, 5] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ()), (1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 20, all discs: True
  Y = [2, 4, 6] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 50, all discs: True
  control Y = [1, 3] (inside a join): lifted regions chi [-3, 1] -> not all discs: True
  control Y = [2, 4] (inside a join): lifted regions chi [-3, 1] -> not all discs: True
  control Y = [1, 2, 3] (inside a join): lifted regions chi [-3, 0, 1] -> not all discs: True
  control Y = [6, 1, 2] (inside a join): lifted regions chi [-3, 0, 1] -> not all discs: True
  control Y = [3, 5] (inside a join): lifted regions chi [-3, 1] -> not all discs: True
  control Y = [4, 6] (inside a join): lifted regions chi [-3, 1] -> not all discs: True
RESULT for this configuration: ALL CHECKS PASSED
====================================================================================================
HEXAGON PICTURE (triangle a,b,c and three spokes tI -> tO): Gamma = C_6
degree d = 7, exponents w = {'a': 1, 'b': 1, 'c': 1, 'tI': 1, 'tO': 3}
cover: chi = -16, genus = 9 ; formula (d-1)(|P|-2)/2 = 9
  X_1 = lift of the disc around ['a', 'c']: connected True, chi -5, boundary curves 1 (gcd formula 1), genus 3, xi 7 ; complement chi [-11]
  X_2 = lift of the disc around ['tI', 'tO']: connected True, chi -5, boundary curves 1 (gcd formula 1), genus 3, xi 7 ; complement chi [-11]
  X_3 = lift of the disc around ['a', 'b']: connected True, chi -5, boundary curves 1 (gcd formula 1), genus 3, xi 7 ; complement chi [-11]
  X_4 = lift of the disc around ['tI', 'tO']: connected True, chi -5, boundary curves 1 (gcd formula 1), genus 3, xi 7 ; complement chi [-11]
  X_5 = lift of the disc around ['b', 'c']: connected True, chi -5, boundary curves 1 (gcd formula 1), genus 3, xi 7 ; complement chi [-11]
  X_6 = lift of the disc around ['tI', 'tO']: connected True, chi -5, boundary curves 1 (gcd formula 1), genus 3, xi 7 ; complement chi [-11]
  pair (1,2) must be DISJOINT: regions (chi, corners) [(-6, 0), (-5, 0), (-5, 0)] -> ok
  pair (1,3) must OVERLAP : regions (chi, corners) [(-5, 14), (1, 14), (1, 14), (1, 14)] -> ok
  pair (1,4) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 14), (1, 14), (1, 14), (1, 14), (1, 28)] -> ok
  pair (1,5) must OVERLAP : regions (chi, corners) [(-5, 14), (1, 14), (1, 14), (1, 14)] -> ok
  pair (1,6) must be DISJOINT: regions (chi, corners) [(-6, 0), (-5, 0), (-5, 0)] -> ok
  pair (2,3) must be DISJOINT: regions (chi, corners) [(-6, 0), (-5, 0), (-5, 0)] -> ok
  pair (2,4) must OVERLAP : regions (chi, corners) [(-5, 14), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 14), (1, 14), (1, 14)] -> ok
  pair (2,5) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 14), (1, 14), (1, 14), (1, 14), (1, 28)] -> ok
  pair (2,6) must OVERLAP : regions (chi, corners) [(-5, 14), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 14), (1, 14), (1, 14)] -> ok
  pair (3,4) must be DISJOINT: regions (chi, corners) [(-6, 0), (-5, 0), (-5, 0)] -> ok
  pair (3,5) must OVERLAP : regions (chi, corners) [(-5, 14), (1, 14), (1, 14), (1, 14)] -> ok
  pair (3,6) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 14), (1, 14), (1, 14), (1, 14), (1, 28)] -> ok
  pair (4,5) must be DISJOINT: regions (chi, corners) [(-6, 0), (-5, 0), (-5, 0)] -> ok
  pair (4,6) must OVERLAP : regions (chi, corners) [(-5, 14), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 14), (1, 14), (1, 14)] -> ok
  pair (5,6) must be DISJOINT: regions (chi, corners) [(-6, 0), (-5, 0), (-5, 0)] -> ok
  Y = [1, 4] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 12, all discs: True
  Y = [2, 5] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 12, all discs: True
  Y = [3, 6] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 12, all discs: True
  Y = [1, 3, 5] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ()), (1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 26, all discs: True
  Y = [2, 4, 6] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 68, all discs: True
  control Y = [1, 3] (inside a join): lifted regions chi [-5, 1] -> not all discs: True
  control Y = [2, 4] (inside a join): lifted regions chi [-5, 1] -> not all discs: True
  control Y = [1, 2, 3] (inside a join): lifted regions chi [-5, 0, 1] -> not all discs: True
  control Y = [6, 1, 2] (inside a join): lifted regions chi [-5, 0, 1] -> not all discs: True
  control Y = [3, 5] (inside a join): lifted regions chi [-5, 1] -> not all discs: True
  control Y = [4, 6] (inside a join): lifted regions chi [-5, 1] -> not all discs: True
RESULT for this configuration: ALL CHECKS PASSED
====================================================================================================
HEXAGON PICTURE (triangle a,b,c and three spokes tI -> tO): Gamma = C_6
degree d = 9, exponents w = {'a': 1, 'b': 1, 'c': 1, 'tI': 2, 'tO': 4}
cover: chi = -22, genus = 12 ; formula (d-1)(|P|-2)/2 = 12
  X_1 = lift of the disc around ['a', 'c']: connected True, chi -7, boundary curves 1 (gcd formula 1), genus 4, xi 10 ; complement chi [-15]
  X_2 = lift of the disc around ['tI', 'tO']: connected True, chi -7, boundary curves 3 (gcd formula 3), genus 3, xi 9 ; complement chi [-15]
  X_3 = lift of the disc around ['a', 'b']: connected True, chi -7, boundary curves 1 (gcd formula 1), genus 4, xi 10 ; complement chi [-15]
  X_4 = lift of the disc around ['tI', 'tO']: connected True, chi -7, boundary curves 3 (gcd formula 3), genus 3, xi 9 ; complement chi [-15]
  X_5 = lift of the disc around ['b', 'c']: connected True, chi -7, boundary curves 1 (gcd formula 1), genus 4, xi 10 ; complement chi [-15]
  X_6 = lift of the disc around ['tI', 'tO']: connected True, chi -7, boundary curves 3 (gcd formula 3), genus 3, xi 9 ; complement chi [-15]
  pair (1,2) must be DISJOINT: regions (chi, corners) [(-8, 0), (-7, 0), (-7, 0)] -> ok
  pair (1,3) must OVERLAP : regions (chi, corners) [(-7, 18), (1, 18), (1, 18), (1, 18)] -> ok
  pair (1,4) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 18), (1, 18), (1, 18), (1, 18), (1, 36)] -> ok
  pair (1,5) must OVERLAP : regions (chi, corners) [(-7, 18), (1, 18), (1, 18), (1, 18)] -> ok
  pair (1,6) must be DISJOINT: regions (chi, corners) [(-8, 0), (-7, 0), (-7, 0)] -> ok
  pair (2,3) must be DISJOINT: regions (chi, corners) [(-8, 0), (-7, 0), (-7, 0)] -> ok
  pair (2,4) must OVERLAP : regions (chi, corners) [(-7, 18), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 18), (1, 18), (1, 18)] -> ok
  pair (2,5) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 18), (1, 18), (1, 18), (1, 18), (1, 36)] -> ok
  pair (2,6) must OVERLAP : regions (chi, corners) [(-7, 18), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 18), (1, 18), (1, 18)] -> ok
  pair (3,4) must be DISJOINT: regions (chi, corners) [(-8, 0), (-7, 0), (-7, 0)] -> ok
  pair (3,5) must OVERLAP : regions (chi, corners) [(-7, 18), (1, 18), (1, 18), (1, 18)] -> ok
  pair (3,6) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 18), (1, 18), (1, 18), (1, 18), (1, 36)] -> ok
  pair (4,5) must be DISJOINT: regions (chi, corners) [(-8, 0), (-7, 0), (-7, 0)] -> ok
  pair (4,6) must OVERLAP : regions (chi, corners) [(-7, 18), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 18), (1, 18), (1, 18)] -> ok
  pair (5,6) must be DISJOINT: regions (chi, corners) [(-8, 0), (-7, 0), (-7, 0)] -> ok
  Y = [1, 4] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 14, all discs: True
  Y = [2, 5] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 14, all discs: True
  Y = [3, 6] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 14, all discs: True
  Y = [1, 3, 5] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ()), (1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 32, all discs: True
  Y = [2, 4, 6] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 86, all discs: True
  control Y = [1, 3] (inside a join): lifted regions chi [-7, 1] -> not all discs: True
  control Y = [2, 4] (inside a join): lifted regions chi [-7, 1] -> not all discs: True
  control Y = [1, 2, 3] (inside a join): lifted regions chi [-7, 0, 1] -> not all discs: True
  control Y = [6, 1, 2] (inside a join): lifted regions chi [-7, 0, 1] -> not all discs: True
  control Y = [3, 5] (inside a join): lifted regions chi [-7, 1] -> not all discs: True
  control Y = [4, 6] (inside a join): lifted regions chi [-7, 1] -> not all discs: True
RESULT for this configuration: ALL CHECKS PASSED
====================================================================================================
HEXAGON PICTURE (triangle a,b,c and three spokes tI -> tO): Gamma = C_6
degree d = 15, exponents w = {'a': 1, 'b': 1, 'c': 1, 'tI': 1, 'tO': 11}
cover: chi = -40, genus = 21 ; formula (d-1)(|P|-2)/2 = 21
  X_1 = lift of the disc around ['a', 'c']: connected True, chi -13, boundary curves 1 (gcd formula 1), genus 7, xi 19 ; complement chi [-27]
  X_2 = lift of the disc around ['tI', 'tO']: connected True, chi -13, boundary curves 3 (gcd formula 3), genus 6, xi 18 ; complement chi [-27]
  X_3 = lift of the disc around ['a', 'b']: connected True, chi -13, boundary curves 1 (gcd formula 1), genus 7, xi 19 ; complement chi [-27]
  X_4 = lift of the disc around ['tI', 'tO']: connected True, chi -13, boundary curves 3 (gcd formula 3), genus 6, xi 18 ; complement chi [-27]
  X_5 = lift of the disc around ['b', 'c']: connected True, chi -13, boundary curves 1 (gcd formula 1), genus 7, xi 19 ; complement chi [-27]
  X_6 = lift of the disc around ['tI', 'tO']: connected True, chi -13, boundary curves 3 (gcd formula 3), genus 6, xi 18 ; complement chi [-27]
  pair (1,2) must be DISJOINT: regions (chi, corners) [(-14, 0), (-13, 0), (-13, 0)] -> ok
  pair (1,3) must OVERLAP : regions (chi, corners) [(-13, 30), (1, 30), (1, 30), (1, 30)] -> ok
  pair (1,4) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 30), (1, 30), (1, 30), (1, 30), (1, 60)] -> ok
  pair (1,5) must OVERLAP : regions (chi, corners) [(-13, 30), (1, 30), (1, 30), (1, 30)] -> ok
  pair (1,6) must be DISJOINT: regions (chi, corners) [(-14, 0), (-13, 0), (-13, 0)] -> ok
  pair (2,3) must be DISJOINT: regions (chi, corners) [(-14, 0), (-13, 0), (-13, 0)] -> ok
  pair (2,4) must OVERLAP : regions (chi, corners) [(-13, 30), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 30), (1, 30), (1, 30)] -> ok
  pair (2,5) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 30), (1, 30), (1, 30), (1, 30), (1, 60)] -> ok
  pair (2,6) must OVERLAP : regions (chi, corners) [(-13, 30), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 30), (1, 30), (1, 30)] -> ok
  pair (3,4) must be DISJOINT: regions (chi, corners) [(-14, 0), (-13, 0), (-13, 0)] -> ok
  pair (3,5) must OVERLAP : regions (chi, corners) [(-13, 30), (1, 30), (1, 30), (1, 30)] -> ok
  pair (3,6) must OVERLAP : regions (chi, corners) [(1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 30), (1, 30), (1, 30), (1, 30), (1, 60)] -> ok
  pair (4,5) must be DISJOINT: regions (chi, corners) [(-14, 0), (-13, 0), (-13, 0)] -> ok
  pair (4,6) must OVERLAP : regions (chi, corners) [(-13, 30), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 4), (1, 30), (1, 30), (1, 30)] -> ok
  pair (5,6) must be DISJOINT: regions (chi, corners) [(-14, 0), (-13, 0), (-13, 0)] -> ok
  Y = [1, 4] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 20, all discs: True
  Y = [2, 5] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 20, all discs: True
  Y = [3, 6] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 20, all discs: True
  Y = [1, 3, 5] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ()), (1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 50, all discs: True
  Y = [2, 4, 6] (not in a join): planar regions (chi, punctures) [(1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ()), (1, ('a',)), (1, ('b',)), (1, ('c',)), (1, ('tI',)), (1, ('tO',))] ; lifted regions 140, all discs: True
  control Y = [1, 3] (inside a join): lifted regions chi [-13, 1] -> not all discs: True
  control Y = [2, 4] (inside a join): lifted regions chi [-13, 1] -> not all discs: True
  control Y = [1, 2, 3] (inside a join): lifted regions chi [-13, 0, 1] -> not all discs: True
  control Y = [6, 1, 2] (inside a join): lifted regions chi [-13, 0, 1] -> not all discs: True
  control Y = [3, 5] (inside a join): lifted regions chi [-13, 1] -> not all discs: True
  control Y = [4, 6] (inside a join): lifted regions chi [-13, 1] -> not all discs: True
RESULT for this configuration: ALL CHECKS PASSED

OVERALL: ALL CONFIGURATIONS PASSED
