$12^{2}_{71}$ - Minimal pinning sets
Pinning sets for 12^2_71 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_71 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 318
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: 3.03847
on average over minimal pinning sets: 2.55
on average over optimal pinning sets: 2.55
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 4, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 4, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {2, 3, 7, 9, 11}
5
[2, 2, 2, 3, 5]
2.80
E (optimal)
• {2, 3, 5, 7, 9}
5
[2, 2, 2, 3, 5]
2.80
F (optimal)
• {2, 3, 7, 9, 10}
5
[2, 2, 2, 3, 4]
2.60
G (optimal)
• {2, 3, 6, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
H (optimal)
• {2, 3, 7, 8, 9}
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
8
0
0
2.55
6
0
0
34
2.77
7
0
0
71
2.94
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
8
0
310
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
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],[1,7,5,2],[3,4,8,9],[3,9,9,7],[4,6,8,8],[5,7,7,9],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[7,16,8,1],[6,9,7,10],[15,8,16,9],[1,11,2,10],[14,5,15,6],[11,5,12,4],[2,17,3,20],[13,19,14,20],[12,19,13,18],[3,17,4,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,8,-2,-9)(11,2,-12,-3)(12,5,-13,-6)(3,6,-4,-7)(7,14,-8,-15)(4,13,-5,-14)(18,9,-19,-10)(10,19,-11,-20)(20,15,-17,-16)(16,17,-1,-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,-9,18)(-2,11,19,9)(-3,-7,-15,20,-11)(-4,-14,7)(-5,12,2,8,14)(-6,3,-12)(-8,1,17,15)(-10,-20,-16,-18)(-13,4,6)(-17,16)(-19,10)(5,13)
Multiloop annotated with half-edges
12^2_71 annotated with half-edges