$12^{2}_{269}$ - Minimal pinning sets
Pinning sets for 12^2_269 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_269 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 480
of which optimal: 10
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12407
on average over minimal pinning sets: 2.72
on average over optimal pinning sets: 2.72
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 2, 5, 8, 11}
5
[2, 2, 3, 3, 4]
2.80
C (optimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {1, 2, 4, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
E (optimal)
• {1, 2, 4, 8, 11}
5
[2, 2, 3, 3, 4]
2.80
F (optimal)
• {1, 2, 4, 5, 9}
5
[2, 2, 3, 3, 3]
2.60
G (optimal)
• {1, 2, 5, 10, 11}
5
[2, 2, 3, 3, 4]
2.80
H (optimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 3, 3, 4]
2.80
I (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
J (optimal)
• {1, 2, 4, 7, 9}
5
[2, 2, 3, 3, 3]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
10
0
0
2.72
6
0
0
53
2.92
7
0
0
118
3.06
8
0
0
143
3.15
9
0
0
102
3.22
10
0
0
43
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
10
0
470
Other information about this multiloop 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],[0,1,6,3],[0,2,7,8],[0,8,8,9],[1,9,6,1],[2,5,9,7],[3,6,9,8],[3,7,4,4],[4,7,6,5]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,6,16,5],[13,4,14,5],[19,3,20,4],[1,10,2,11],[6,17,7,16],[7,12,8,13],[8,18,9,19],[9,2,10,3],[11,18,12,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,14,-8,-1)(18,1,-19,-2)(11,2,-12,-3)(3,20,-4,-15)(13,8,-14,-9)(5,10,-6,-11)(16,9,-17,-10)(6,17,-7,-18)(12,19,-13,-20)(15,4,-16,-5)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-7)(-2,11,-6,-18)(-3,-15,-5,-11)(-4,15)(-8,13,19,1)(-9,16,4,20,-13)(-10,5,-16)(-12,-20,3)(-14,7,17,9)(-17,6,10)(-19,12,2)(8,14)
Multiloop annotated with half-edges
12^2_269 annotated with half-edges