$12^{1}_{181}$ - Minimal pinning sets
Pinning sets for 12^1_181 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_181 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 360
of which optimal: 4
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10609
on average over minimal pinning sets: 2.67619
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 10, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 2, 5, 10, 11}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 2, 7, 10, 11}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 2, 7, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 3, 6, 7, 10, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
b (minimal)
• {1, 3, 6, 7, 9, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
c (minimal)
• {1, 3, 5, 7, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.6
6
0
3
25
2.82
7
0
0
77
2.99
8
0
0
110
3.12
9
0
0
89
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
4
3
353
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,8,8],[0,9,6,5],[1,4,2,1],[2,4,9,7],[3,6,9,8],[3,7,9,3],[4,8,7,6]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[10,17,11,18],[19,16,20,17],[12,6,13,5],[1,8,2,9],[18,9,19,10],[2,15,3,16],[6,3,7,4],[13,4,14,5],[14,7,15,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,7,-1,-8)(9,2,-10,-3)(16,3,-17,-4)(13,4,-14,-5)(17,10,-18,-11)(14,11,-15,-12)(5,12,-6,-13)(6,15,-7,-16)(1,18,-2,-19)(8,19,-9,-20)
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)(-2,9,19)(-3,16,-7,20,-9)(-4,13,-6,-16)(-5,-13)(-8,-20)(-10,17,3)(-11,14,4,-17)(-12,5,-14)(-15,6,12)(-18,1,7,15,11)(2,18,10)
Loop annotated with half-edges
12^1_181 annotated with half-edges