$12^{2}_{319}$ - Minimal pinning sets
Pinning sets for 12^2_319 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_319 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 460
of which optimal: 9
of which minimal: 11
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12087
on average over minimal pinning sets: 2.77662
on average over optimal pinning sets: 2.71111
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 5, 9}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 2, 4, 5, 9}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 2, 4, 7, 10}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {1, 2, 4, 9, 10}
5
[2, 2, 3, 3, 3]
2.60
E (optimal)
• {1, 2, 3, 4, 10}
5
[2, 2, 3, 3, 3]
2.60
F (optimal)
• {1, 3, 4, 6, 10}
5
[2, 2, 3, 3, 4]
2.80
G (optimal)
• {1, 2, 4, 6, 10}
5
[2, 2, 3, 3, 4]
2.80
H (optimal)
• {1, 2, 4, 10, 12}
5
[2, 2, 3, 3, 5]
3.00
I (optimal)
• {1, 3, 4, 5, 10}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 4, 8, 9, 11}
6
[2, 2, 3, 3, 4, 4]
3.00
b (minimal)
• {1, 3, 4, 6, 8, 9, 11}
7
[2, 2, 3, 3, 4, 4, 4]
3.14
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
9
0
0
2.71
6
0
1
47
2.9
7
0
1
109
3.04
8
0
0
138
3.15
9
0
0
101
3.22
10
0
0
43
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
9
2
449
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],[0,5,6,7],[0,8,9,3],[0,2,6,4],[0,3,5,5],[1,4,4,6],[1,5,3,9],[1,9,8,8],[2,7,7,9],[2,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,3,18,4],[6,15,7,16],[7,19,8,20],[1,8,2,9],[9,2,10,3],[18,10,19,11],[4,14,5,13],[5,12,6,13],[14,11,15,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,4,-14,-5)(17,6,-18,-7)(10,7,-11,-8)(1,8,-2,-9)(9,16,-10,-1)(2,11,-3,-12)(5,14,-6,-15)(20,15,-17,-16)(3,18,-4,-19)(12,19,-13,-20)
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)(-2,-12,-20,-16,9)(-3,-19,12)(-4,13,19)(-5,-15,20,-13)(-6,17,15)(-7,10,16,-17)(-8,1,-10)(-11,2,8)(-14,5)(-18,3,11,7)(4,18,6,14)
Multiloop annotated with half-edges
12^2_319 annotated with half-edges