$12^{3}_{78}$ - Minimal pinning sets
Pinning sets for 12^3_78 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_78 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 400
of which optimal: 10
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05663
on average over minimal pinning sets: 2.56
on average over optimal pinning sets: 2.56
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 8}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 6, 10}
5
[2, 2, 2, 4, 4]
2.80
C (optimal)
• {1, 2, 3, 6, 8}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 2, 3, 5, 8}
5
[2, 2, 2, 3, 3]
2.40
E (optimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 2, 3, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
G (optimal)
• {1, 2, 3, 7, 11}
5
[2, 2, 2, 3, 5]
2.80
H (optimal)
• {1, 2, 3, 6, 11}
5
[2, 2, 2, 3, 4]
2.60
I (optimal)
• {1, 2, 3, 5, 11}
5
[2, 2, 2, 3, 3]
2.40
J (optimal)
• {1, 2, 3, 8, 12}
5
[2, 2, 2, 3, 5]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
10
0
0
2.56
6
0
0
48
2.81
7
0
0
99
2.98
8
0
0
115
3.1
9
0
0
82
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
10
0
390
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: [[1,2,3,4],[0,4,5,6],[0,7,7,3],[0,2,5,8],[0,8,8,1],[1,8,3,9],[1,9,9,7],[2,6,9,2],[3,5,4,4],[5,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[10,14,1,11],[11,15,12,20],[9,4,10,5],[13,3,14,4],[1,16,2,15],[12,17,13,18],[7,19,8,20],[5,8,6,9],[2,16,3,17],[18,6,19,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-17,-2)(7,2,-8,-3)(13,4,-14,-5)(3,6,-4,-7)(8,17,-9,-18)(5,14,-6,-15)(15,18,-16,-19)(16,9,-11,-10)(10,11,-1,-12)(19,12,-20,-13)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,12)(-2,7,-4,13,-20)(-3,-7)(-5,-15,-19,-13)(-6,3,-8,-18,15)(-9,16,18)(-10,-12,19,-16)(-11,10)(-14,5)(-17,8,2)(1,11,9,17)(4,6,14)
Multiloop annotated with half-edges
12^3_78 annotated with half-edges