$12^{2}_{268}$ - Minimal pinning sets
Pinning sets for 12^2_268 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_268 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 603
of which optimal: 16
of which minimal: 18
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.14136
on average over minimal pinning sets: 2.84074
on average over optimal pinning sets: 2.8
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, 3, 3, 4]
2.80
B (optimal)
• {1, 4, 7, 9, 12}
5
[2, 2, 3, 3, 4]
2.80
C (optimal)
• {1, 4, 7, 9, 11}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {1, 4, 7, 8, 11}
5
[2, 2, 3, 4, 4]
3.00
E (optimal)
• {1, 2, 4, 7, 9}
5
[2, 2, 3, 3, 3]
2.60
F (optimal)
• {2, 4, 5, 7, 10}
5
[2, 2, 3, 3, 4]
2.80
G (optimal)
• {2, 4, 5, 7, 9}
5
[2, 2, 3, 3, 3]
2.60
H (optimal)
• {1, 4, 5, 7, 9}
5
[2, 2, 3, 3, 3]
2.60
I (optimal)
• {1, 4, 5, 7, 8}
5
[2, 2, 3, 3, 4]
2.80
J (optimal)
• {1, 4, 5, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
K (optimal)
• {1, 2, 4, 5, 7}
5
[2, 2, 3, 3, 3]
2.60
L (optimal)
• {2, 4, 6, 7, 9}
5
[2, 2, 3, 3, 4]
2.80
M (optimal)
• {2, 4, 6, 7, 11}
5
[2, 2, 3, 4, 4]
3.00
N (optimal)
• {2, 3, 4, 6, 7}
5
[2, 2, 3, 4, 4]
3.00
O (optimal)
• {1, 4, 6, 7, 11}
5
[2, 2, 3, 4, 4]
3.00
P (optimal)
• {2, 4, 5, 6, 7}
5
[2, 2, 3, 3, 4]
2.80
a (minimal)
• {2, 3, 4, 7, 8, 10}
6
[2, 2, 3, 4, 4, 4]
3.17
b (minimal)
• {1, 3, 4, 7, 8, 12}
6
[2, 2, 3, 4, 4, 4]
3.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
16
0
0
2.8
6
0
2
78
2.98
7
0
0
164
3.1
8
0
0
176
3.18
9
0
0
112
3.24
10
0
0
44
3.28
11
0
0
10
3.31
12
0
0
1
3.33
Total
16
2
585
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,7,3],[0,2,8,9],[0,9,9,5],[1,4,8,6],[1,5,7,7],[2,6,6,8],[3,7,5,9],[3,8,4,4]]
PD code (use to draw this multiloop with SnapPy): [[14,9,1,10],[10,6,11,5],[13,4,14,5],[8,3,9,4],[1,15,2,20],[6,20,7,19],[11,19,12,18],[12,17,13,18],[7,16,8,17],[2,15,3,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(11,2,-12,-3)(3,8,-4,-9)(12,7,-13,-8)(5,16,-6,-17)(10,17,-11,-18)(18,9,-19,-10)(19,4,-20,-5)(20,13,-15,-14)(14,15,-1,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,16)(-2,11,17,-6)(-3,-9,18,-11)(-4,19,9)(-5,-17,10,-19)(-7,12,2)(-8,3,-12)(-10,-18)(-13,20,4,8)(-14,-16,5,-20)(-15,14)(1,15,13,7)
Multiloop annotated with half-edges
12^2_268 annotated with half-edges