$12^{3}_{9}$ - Minimal pinning sets
Pinning sets for 12^3_9 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_9 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 300
of which optimal: 4
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04512
on average over minimal pinning sets: 2.54167
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, 4, 5, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {2, 4, 5, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {2, 4, 5, 9, 12}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {2, 4, 5, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 2, 3, 5, 7, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 5, 8, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 2, 3, 5, 9, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
d (minimal)
• {1, 2, 3, 5, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.5
6
0
4
22
2.73
7
0
0
66
2.92
8
0
0
89
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
4
4
292
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: [[1,2,3,4],[0,4,5,6],[0,7,7,8],[0,8,9,9],[0,5,5,1],[1,4,4,6],[1,5,9,7],[2,6,8,2],[2,7,9,3],[3,8,6,3]]
PD code (use to draw this multiloop with SnapPy): [[6,14,1,7],[7,10,8,11],[5,20,6,15],[17,13,18,14],[1,9,2,10],[8,2,9,3],[11,3,12,4],[15,4,16,5],[16,19,17,20],[12,18,13,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(16,1,-17,-2)(12,5,-13,-6)(4,13,-5,-14)(11,14,-12,-7)(18,9,-19,-10)(3,10,-4,-11)(8,19,-9,-20)(17,20,-18,-15)(2,15,-3,-16)
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,-3,-11,-7)(-2,-16)(-4,-14,11)(-5,12,14)(-6,7,-12)(-8,-20,17,1)(-9,18,20)(-10,3,15,-18)(-13,4,10,-19,8,6)(-15,2,-17)(5,13)(9,19)
Multiloop annotated with half-edges
12^3_9 annotated with half-edges