$12^{2}_{338}$ - Minimal pinning sets
Pinning sets for 12^2_338 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_338 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 192
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: 2.99376
on average over minimal pinning sets: 2.45833
on average over optimal pinning sets: 2.45833
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 9, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 4, 9, 10}
6
[2, 2, 2, 2, 3, 4]
2.50
C (optimal)
• {1, 2, 3, 4, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 2, 3, 4, 7, 11}
6
[2, 2, 2, 2, 3, 4]
2.50
E (optimal)
• {1, 2, 3, 4, 7, 10}
6
[2, 2, 2, 2, 4, 4]
2.67
F (optimal)
• {1, 2, 3, 4, 7, 8}
6
[2, 2, 2, 2, 3, 4]
2.50
G (optimal)
• {1, 2, 3, 4, 5, 10}
6
[2, 2, 2, 2, 3, 4]
2.50
H (optimal)
• {1, 2, 3, 4, 5, 8}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
8
0
0
2.46
7
0
0
34
2.76
8
0
0
59
2.96
9
0
0
54
3.1
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
8
0
184
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,6,3],[0,2,7,7],[0,8,8,1],[1,8,9,6],[1,5,2,2],[3,9,9,3],[4,9,5,4],[5,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,11,16,12],[13,6,14,7],[19,5,20,6],[1,10,2,11],[16,9,17,8],[12,8,13,7],[4,18,5,19],[9,2,10,3],[17,3,18,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(9,2,-10,-3)(3,8,-4,-9)(4,19,-5,-20)(11,6,-12,-7)(18,7,-19,-8)(5,12,-6,-13)(20,13,-15,-14)(14,15,-1,-16)(10,17,-11,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16)(-2,9,-4,-20,-14,-16)(-3,-9)(-5,-13,20)(-6,11,17,1,15,13)(-7,18,-11)(-8,3,-10,-18)(-12,5,19,7)(-15,14)(-17,10,2)(-19,4,8)(6,12)
Multiloop annotated with half-edges
12^2_338 annotated with half-edges