$12^{2}_{332}$ - Minimal pinning sets
Pinning sets for 12^2_332 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_332 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 659
of which optimal: 11
of which minimal: 19
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.21944
on average over minimal pinning sets: 3.0193
on average over optimal pinning sets: 3.01818
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 7, 8, 12}
5
[2, 3, 3, 3, 4]
3.00
B (optimal)
• {1, 2, 7, 10, 12}
5
[2, 3, 3, 4, 4]
3.20
C (optimal)
• {1, 2, 7, 9, 11}
5
[2, 3, 3, 3, 3]
2.80
D (optimal)
• {1, 2, 7, 11, 12}
5
[2, 3, 3, 3, 4]
3.00
E (optimal)
• {1, 4, 5, 7, 10}
5
[2, 3, 3, 3, 4]
3.00
F (optimal)
• {1, 4, 7, 10, 12}
5
[2, 3, 3, 4, 4]
3.20
G (optimal)
• {1, 4, 7, 9, 11}
5
[2, 3, 3, 3, 3]
2.80
H (optimal)
• {1, 4, 7, 10, 11}
5
[2, 3, 3, 3, 4]
3.00
I (optimal)
• {1, 2, 4, 7, 11}
5
[2, 3, 3, 3, 3]
2.80
J (optimal)
• {1, 3, 5, 8, 9}
5
[2, 3, 3, 3, 4]
3.00
K (optimal)
• {1, 3, 7, 10, 12}
5
[2, 3, 4, 4, 4]
3.40
a (minimal)
• {1, 2, 5, 8, 9, 12}
6
[2, 3, 3, 3, 3, 4]
3.00
b (minimal)
• {1, 2, 5, 8, 9, 11}
6
[2, 3, 3, 3, 3, 3]
2.83
c (minimal)
• {1, 4, 5, 8, 9, 10}
6
[2, 3, 3, 3, 3, 4]
3.00
d (minimal)
• {1, 4, 5, 8, 9, 11}
6
[2, 3, 3, 3, 3, 3]
2.83
e (minimal)
• {1, 2, 4, 5, 8, 9}
6
[2, 3, 3, 3, 3, 3]
2.83
f (minimal)
• {1, 3, 7, 8, 9, 12}
6
[2, 3, 3, 3, 4, 4]
3.17
g (minimal)
• {1, 3, 5, 7, 9, 10}
6
[2, 3, 3, 3, 4, 4]
3.17
h (minimal)
• {1, 3, 6, 7, 9, 11}
6
[2, 3, 3, 3, 4, 5]
3.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
11
0
0
3.02
6
0
8
62
3.1
7
0
0
172
3.18
8
0
0
206
3.24
9
0
0
136
3.28
10
0
0
52
3.3
11
0
0
11
3.32
12
0
0
1
3.33
Total
11
8
640
Other information about this multiloop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,3],[0,2,8,4],[0,3,8,1],[1,8,9,6],[1,5,7,2],[2,6,9,9],[3,9,5,4],[5,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,13,8,12],[5,15,6,16],[19,14,20,15],[1,14,2,13],[8,18,9,17],[11,16,12,17],[4,10,5,11],[18,2,19,3],[9,3,10,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(16,5,-17,-6)(1,8,-2,-9)(10,19,-11,-20)(11,4,-12,-5)(17,12,-18,-13)(2,13,-3,-14)(9,14,-10,-15)(15,20,-16,-7)(3,18,-4,-19)
Edge permutation $\epsilon=$ (-1,1)(-2,2)(-3,3)(-4,4)(-5,5)(-6,6)(-7,7)(-8,8)(-9,9)(-10,10)(-11,11)(-12,12)(-13,13)(-14,14)(-15,15)(-16,16)(-17,17)(-18,18)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-9,-15,-7)(-2,-14,9)(-3,-19,10,14)(-4,11,19)(-5,16,20,-11)(-6,7,-16)(-8,1)(-10,-20,15)(-12,17,5)(-13,2,8,6,-17)(-18,3,13)(4,18,12)
Multiloop annotated with half-edges
12^2_332 annotated with half-edges