$12^{2}_{167}$ - Minimal pinning sets
Pinning sets for 12^2_167 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_167 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 8, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 4, 5, 9, 12}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,6,6,7],[0,7,8,8],[1,9,9,5],[1,4,6,6],[2,5,5,2],[2,9,8,3],[3,7,9,3],[4,8,7,4]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,5,8,6],[10,19,11,20],[1,14,2,13],[16,4,17,5],[8,17,9,18],[18,9,19,10],[11,15,12,14],[2,12,3,13],[3,15,4,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(13,2,-14,-3)(10,15,-11,-16)(5,18,-6,-19)(19,4,-20,-5)(20,17,-7,-18)(6,7,-1,-8)(14,9,-15,-10)(16,11,-17,-12)(3,12,-4,-13)
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)(-2,13,-4,19,-6,-8)(-3,-13)(-5,-19)(-7,6,18)(-9,14,2)(-10,-16,-12,3,-14)(-11,16)(-15,10)(-17,20,4,12)(-18,5,-20)(1,7,17,11,15,9)
Multiloop annotated with half-edges
12^2_167 annotated with half-edges