$12^{1}_{22}$ - Minimal pinning sets
Pinning sets for 12^1_22 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_22 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 384
of which optimal: 6
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.1089
on average over minimal pinning sets: 2.66667
on average over optimal pinning sets: 2.66667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 7, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {2, 3, 6, 7, 9}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {2, 3, 5, 7, 8}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {2, 3, 7, 8, 12}
5
[2, 2, 3, 3, 4]
2.80
E (optimal)
• {1, 2, 4, 7, 8}
5
[2, 2, 3, 3, 3]
2.60
F (optimal)
• {1, 2, 3, 7, 8}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 4, 6, 7, 9}
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
6
0
0
2.67
6
0
1
34
2.87
7
0
0
86
3.01
8
0
0
115
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
6
1
377
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],[0,4,5,6],[0,6,7,8],[0,8,9,6],[0,5,5,1],[1,4,4,6],[1,5,3,2],[2,9,9,8],[2,7,9,3],[3,8,7,7]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[14,11,15,12],[4,19,5,20],[16,9,17,10],[1,13,2,12],[2,13,3,14],[3,10,4,11],[7,18,8,19],[5,8,6,9],[17,6,18,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,2,-14,-3)(1,4,-2,-5)(12,5,-13,-6)(19,6,-20,-7)(10,7,-11,-8)(8,17,-9,-18)(18,9,-19,-10)(3,14,-4,-15)(20,15,-1,-16)(11,16,-12,-17)
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,-5,12,16)(-2,13,5)(-3,-15,20,6,-13)(-4,1,15)(-6,19,9,17,-12)(-7,10,-19)(-8,-18,-10)(-9,18)(-11,-17,8)(-14,3)(-16,11,7,-20)(2,4,14)
Loop annotated with half-edges
12^1_22 annotated with half-edges