$12^{1}_{106}$ - Minimal pinning sets
Pinning sets for 12^1_106 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_106 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 376
of which optimal: 5
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10975
on average over minimal pinning sets: 2.67143
on average over optimal pinning sets: 2.64
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 4, 5, 11, 12}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 4, 5, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 4, 5, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
E (optimal)
• {1, 3, 4, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 3, 5, 11, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 2, 3, 5, 7, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
5
0
0
2.64
6
0
2
30
2.85
7
0
0
83
3.01
8
0
0
114
3.12
9
0
0
90
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
5
2
369
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,6,7],[0,5,1,1],[1,4,8,2],[2,9,7,3],[3,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,9,3,10],[12,19,13,20],[4,13,5,14],[1,11,2,10],[11,8,12,9],[18,5,19,6],[14,18,15,17],[7,16,8,17],[6,16,7,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,3,-17,-4)(9,6,-10,-7)(20,7,-1,-8)(8,19,-9,-20)(2,11,-3,-12)(13,4,-14,-5)(5,14,-6,-15)(15,12,-16,-13)(10,17,-11,-18)(1,18,-2,-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,-19,8)(-2,-12,15,-6,9,19)(-3,16,12)(-4,13,-16)(-5,-15,-13)(-7,20,-9)(-8,-20)(-10,-18,1,7)(-11,2,18)(-14,5)(-17,10,6,14,4)(3,11,17)
Loop annotated with half-edges
12^1_106 annotated with half-edges