$12^{2}_{291}$ - Minimal pinning sets
Pinning sets for 12^2_291 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_291 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 384
of which optimal: 6
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11498
on average over minimal pinning sets: 2.69167
on average over optimal pinning sets: 2.7
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 7, 9}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 3, 5, 7, 9}
5
[2, 2, 3, 3, 4]
2.80
C (optimal)
• {1, 5, 7, 9, 12}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {1, 5, 7, 9, 10}
5
[2, 2, 3, 3, 3]
2.60
E (optimal)
• {1, 5, 7, 10, 11}
5
[2, 2, 3, 3, 4]
2.80
F (optimal)
• {1, 2, 5, 7, 10}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 4, 6, 7, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
b (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
2.7
6
0
2
33
2.89
7
0
0
86
3.02
8
0
0
115
3.13
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
2
376
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,5,6,0],[0,7,4,1],[1,3,7,5],[1,4,8,2],[2,8,9,9],[3,9,8,4],[5,7,9,6],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[12,5,1,6],[6,13,7,20],[4,11,5,12],[1,14,2,13],[7,2,8,3],[3,19,4,20],[16,10,17,11],[14,9,15,8],[15,18,16,19],[9,17,10,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,12,-6,-1)(10,3,-11,-4)(6,15,-7,-16)(17,8,-18,-9)(2,9,-3,-10)(7,18,-8,-19)(16,19,-17,-20)(1,20,-2,-13)(13,4,-14,-5)(14,11,-15,-12)
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,-13,-5)(-2,-10,-4,13)(-3,10)(-6,-16,-20,1)(-7,-19,16)(-8,17,19)(-9,2,20,-17)(-11,14,4)(-12,5,-14)(-15,6,12)(-18,7,15,11,3,9)(8,18)
Multiloop annotated with half-edges
12^2_291 annotated with half-edges