$12^{2}_{66}$ - Minimal pinning sets
Pinning sets for 12^2_66 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_66 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 312
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04121
on average over minimal pinning sets: 2.5
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 4, 6, 8}
5
[2, 2, 2, 3, 4]
2.60
B (optimal)
• {2, 3, 5, 6, 8}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {2, 3, 6, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {2, 3, 6, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
E (optimal)
• {2, 3, 6, 8, 9}
5
[2, 2, 2, 3, 5]
2.80
F (optimal)
• {1, 2, 3, 6, 8}
5
[2, 2, 2, 3, 3]
2.40
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
2.5
6
0
0
31
2.76
7
0
0
70
2.93
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
6
0
306
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,2],[0,1,4,0],[0,5,6,1],[1,7,8,2],[3,8,8,6],[3,5,9,7],[4,6,9,9],[4,9,5,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,18,14,19],[19,11,20,12],[1,17,2,18],[14,10,15,11],[16,6,17,7],[2,6,3,5],[9,4,10,5],[15,8,16,7],[3,8,4,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,12,-6,-1)(1,10,-2,-11)(13,2,-14,-3)(3,20,-4,-13)(11,4,-12,-5)(19,6,-20,-7)(16,7,-17,-8)(9,14,-10,-15)(18,15,-19,-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,-11,-5)(-2,13,-4,11)(-3,-13)(-6,19,15,-10,1)(-7,16,-19)(-8,-18,-16)(-9,-15,18)(-12,5)(-14,9,17,7,-20,3)(-17,8)(2,10,14)(4,20,6,12)
Multiloop annotated with half-edges
12^2_66 annotated with half-edges