$12^{2}_{264}$ - Minimal pinning sets
Pinning sets for 12^2_264 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_264 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 574
of which optimal: 7
of which minimal: 19
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.20319
on average over minimal pinning sets: 2.97018
on average over optimal pinning sets: 2.94286
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {3, 5, 6, 10, 11}
5
[2, 3, 3, 3, 4]
3.00
B (optimal)
• {3, 4, 5, 9, 12}
5
[2, 3, 3, 3, 4]
3.00
C (optimal)
• {1, 2, 5, 9, 12}
5
[2, 3, 3, 3, 3]
2.80
D (optimal)
• {2, 5, 7, 10, 12}
5
[2, 3, 3, 3, 4]
3.00
E (optimal)
• {2, 5, 7, 9, 12}
5
[2, 3, 3, 3, 4]
3.00
F (optimal)
• {2, 5, 6, 9, 12}
5
[2, 3, 3, 3, 3]
2.80
G (optimal)
• {2, 4, 5, 9, 12}
5
[2, 3, 3, 3, 4]
3.00
a (minimal)
• {1, 3, 5, 6, 9, 12}
6
[2, 3, 3, 3, 3, 3]
2.83
b (minimal)
• {3, 5, 6, 9, 11, 12}
6
[2, 3, 3, 3, 3, 4]
3.00
c (minimal)
• {1, 3, 5, 6, 9, 10}
6
[2, 3, 3, 3, 3, 3]
2.83
d (minimal)
• {1, 3, 4, 5, 9, 10}
6
[2, 3, 3, 3, 3, 4]
3.00
e (minimal)
• {3, 4, 5, 9, 10, 11}
6
[2, 3, 3, 3, 4, 4]
3.17
f (minimal)
• {3, 4, 5, 7, 10, 11}
6
[2, 3, 3, 4, 4, 4]
3.33
g (minimal)
• {1, 2, 5, 6, 10, 12}
6
[2, 3, 3, 3, 3, 3]
2.83
h (minimal)
• {2, 5, 6, 10, 11, 12}
6
[2, 3, 3, 3, 3, 4]
3.00
i (minimal)
• {1, 2, 3, 5, 9, 10}
6
[2, 3, 3, 3, 3, 3]
2.83
j (minimal)
• {1, 2, 3, 5, 7, 10}
6
[2, 3, 3, 3, 3, 4]
3.00
k (minimal)
• {2, 3, 5, 7, 10, 11}
6
[2, 3, 3, 3, 4, 4]
3.17
l (minimal)
• {1, 2, 3, 5, 6, 10}
6
[2, 3, 3, 3, 3, 3]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
7
0
0
2.94
6
0
12
41
3.05
7
0
0
141
3.15
8
0
0
182
3.22
9
0
0
128
3.27
10
0
0
51
3.3
11
0
0
11
3.32
12
0
0
1
3.33
Total
7
12
555
Other information about this multiloop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,6],[0,6,7,3],[0,2,8,0],[1,8,9,5],[1,4,9,6],[1,5,7,2],[2,6,9,8],[3,7,9,4],[4,8,7,5]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,10,16,9],[13,2,14,3],[19,1,20,2],[10,6,11,5],[16,5,17,4],[8,3,9,4],[12,7,13,8],[18,6,19,7],[11,18,12,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,14,-16,-1)(6,1,-7,-2)(3,18,-4,-19)(16,9,-17,-10)(7,10,-8,-11)(2,11,-3,-12)(12,19,-13,-20)(13,4,-14,-5)(8,17,-9,-18)(20,5,-15,-6)
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,-15)(-2,-12,-20,-6)(-3,-19,12)(-4,13,19)(-5,20,-13)(-7,-11,2)(-8,-18,3,11)(-9,16,14,4,18)(-10,7,1,-16)(-14,15,5)(-17,8,10)(9,17)
Multiloop annotated with half-edges
12^2_264 annotated with half-edges