$12^{2}_{214}$ - Minimal pinning sets
Pinning sets for 12^2_214 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_214 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 276
of which optimal: 2
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04954
on average over minimal pinning sets: 2.5625
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 12}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 8, 12}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 2, 3, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 5, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 2, 3, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
d (minimal)
• {1, 2, 3, 8, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
e (minimal)
• {1, 2, 3, 7, 9, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
f (minimal)
• {1, 2, 3, 5, 9, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.5
6
0
6
13
2.69
7
0
0
57
2.9
8
0
0
84
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
6
268
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,7],[0,7,8,4],[0,3,9,9],[1,9,6,6],[1,5,5,2],[2,8,8,3],[3,7,7,9],[4,8,5,4]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,10,8,11],[11,5,12,6],[19,16,20,17],[1,16,2,15],[3,9,4,10],[8,4,9,5],[12,18,13,17],[13,18,14,19],[2,14,3,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-7,-2)(9,2,-10,-3)(6,7,-1,-8)(19,8,-20,-9)(5,12,-6,-13)(13,4,-14,-5)(17,14,-18,-15)(15,10,-16,-11)(11,16,-12,-17)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,8)(-2,9,-20)(-3,-19,-9)(-4,13,-6,-8,19)(-5,-13)(-7,6,12,16,10,2)(-10,15,-18,3)(-11,-17,-15)(-12,5,-14,17)(-16,11)(1,7)(4,18,14)
Multiloop annotated with half-edges
12^2_214 annotated with half-edges