$12^{1}_{328}$ - Minimal pinning sets
Pinning sets for 12^1_328 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_328 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 430
of which optimal: 11
of which minimal: 12
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.06287
on average over minimal pinning sets: 2.61944
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, 2, 3, 7, 10}
5
[2, 2, 2, 4, 4]
2.80
B (optimal)
• {1, 2, 3, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 3, 7, 8}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 2, 3, 9, 12}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {1, 2, 3, 7, 12}
5
[2, 2, 2, 4, 4]
2.80
F (optimal)
• {1, 2, 3, 4, 9}
5
[2, 2, 2, 3, 3]
2.40
G (optimal)
• {1, 2, 3, 4, 10}
5
[2, 2, 2, 3, 4]
2.60
H (optimal)
• {1, 2, 3, 4, 7}
5
[2, 2, 2, 3, 4]
2.60
I (optimal)
• {1, 2, 3, 4, 8}
5
[2, 2, 2, 3, 3]
2.40
J (optimal)
• {1, 2, 3, 6, 10}
5
[2, 2, 2, 4, 4]
2.80
K (optimal)
• {1, 2, 3, 6, 8}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 2, 3, 5, 9, 10}
6
[2, 2, 2, 3, 4, 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
11
0
0
2.6
6
0
1
54
2.84
7
0
0
111
3.0
8
0
0
123
3.11
9
0
0
84
3.19
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
11
1
418
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 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],[0,1,6,7],[0,7,7,4],[0,3,8,5],[1,4,6,1],[2,5,8,9],[2,9,3,3],[4,9,9,6],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,13,9,14],[19,12,20,13],[10,5,11,6],[1,6,2,7],[14,7,15,8],[15,18,16,19],[4,11,5,12],[2,17,3,18],[16,3,17,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(8,3,-9,-4)(19,4,-20,-5)(12,5,-13,-6)(2,9,-3,-10)(17,10,-18,-11)(6,11,-7,-12)(20,15,-1,-16)(13,16,-14,-17)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,16)(-2,-10,17,-14)(-3,8,18,10)(-4,19,-8)(-5,12,-7,-19)(-6,-12)(-9,2,-15,20,4)(-11,6,-13,-17)(-16,13,5,-20)(-18,7,11)(1,15)(3,9)
Loop annotated with half-edges
12^1_328 annotated with half-edges