$12^{1}_{171}$ - Minimal pinning sets
Pinning sets for 12^1_171 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_171 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 432
of which optimal: 8
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11442
on average over minimal pinning sets: 2.675
on average over optimal pinning sets: 2.675
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 5, 8}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 3, 4, 5, 6}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 2, 5, 6, 8}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 2, 5, 8, 12}
5
[2, 2, 3, 3, 4]
2.80
E (optimal)
• {1, 2, 5, 8, 11}
5
[2, 2, 3, 3, 4]
2.80
F (optimal)
• {1, 2, 3, 5, 8}
5
[2, 2, 3, 3, 3]
2.60
G (optimal)
• {1, 2, 4, 5, 6}
5
[2, 2, 3, 3, 3]
2.60
H (optimal)
• {1, 2, 5, 6, 7}
5
[2, 2, 3, 3, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
8
0
0
2.68
6
0
0
44
2.89
7
0
0
102
3.03
8
0
0
129
3.14
9
0
0
96
3.22
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
8
0
424
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 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],[0,6,7,8],[0,9,6,5],[1,4,2,1],[2,4,7,3],[3,6,9,8],[3,7,9,9],[4,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,17,9,18],[19,16,20,17],[10,3,11,4],[1,6,2,7],[18,7,19,8],[2,15,3,16],[11,15,12,14],[4,14,5,13],[5,12,6,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(19,4,-20,-5)(16,5,-17,-6)(3,10,-4,-11)(11,2,-12,-3)(12,9,-13,-10)(20,13,-1,-14)(17,14,-18,-15)(6,15,-7,-16)(7,18,-8,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,18,14)(-2,11,-4,19,-8)(-3,-11)(-5,16,-7,-19)(-6,-16)(-9,12,2)(-10,3,-12)(-13,20,4,10)(-14,17,5,-20)(-15,6,-17)(-18,7,15)(1,13,9)
Loop annotated with half-edges
12^1_171 annotated with half-edges