$12^{2}_{266}$ - Minimal pinning sets
Pinning sets for 12^2_266 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_266 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 512
of which optimal: 2
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12658
on average over minimal pinning sets: 2.70333
on average over optimal pinning sets: 2.625
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 9}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 2, 7, 12}
4
[2, 2, 3, 4]
2.75
a (minimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 2, 6, 7, 9, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
c (minimal)
• {1, 2, 4, 5, 7, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.62
5
0
1
16
2.82
6
0
2
61
2.96
7
0
0
126
3.07
8
0
0
147
3.16
9
0
0
103
3.23
10
0
0
43
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
3
507
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],[0,5,6,2],[0,1,6,3],[0,2,7,8],[0,8,8,5],[1,4,9,9],[1,9,7,2],[3,6,9,8],[3,7,4,4],[5,7,6,5]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,6,12,5],[9,4,10,5],[19,3,20,4],[1,15,2,16],[6,16,7,17],[12,8,13,9],[13,18,14,19],[14,2,15,3],[7,18,8,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,10,-12,-1)(16,1,-17,-2)(7,2,-8,-3)(3,18,-4,-19)(19,6,-20,-7)(9,12,-10,-13)(4,13,-5,-14)(14,5,-15,-6)(20,15,-11,-16)(8,17,-9,-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,-11)(-2,7,-20,-16)(-3,-19,-7)(-4,-14,-6,19)(-5,14)(-8,-18,3)(-9,-13,4,18)(-10,11,15,5,13)(-12,9,17,1)(-15,20,6)(-17,8,2)(10,12)
Multiloop annotated with half-edges
12^2_266 annotated with half-edges