$12^{3}_{69}$ - Minimal pinning sets
Pinning sets for 12^3_69 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_69 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 747
of which optimal: 2
of which minimal: 25
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.15174
on average over minimal pinning sets: 2.86667
on average over optimal pinning sets: 3.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 7, 11}
4
[2, 2, 3, 5]
3.00
B (optimal)
• {1, 2, 8, 12}
4
[2, 2, 3, 5]
3.00
a (minimal)
• {1, 2, 3, 4, 8}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 2, 4, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 2, 6, 9, 10}
5
[2, 2, 3, 4, 4]
3.00
d (minimal)
• {1, 2, 6, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
e (minimal)
• {1, 2, 6, 7, 10}
5
[2, 2, 4, 4, 5]
3.40
f (minimal)
• {1, 2, 3, 5, 8}
5
[2, 2, 3, 3, 3]
2.60
g (minimal)
• {1, 2, 5, 8, 10}
5
[2, 2, 3, 3, 4]
2.80
h (minimal)
• {1, 2, 5, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
i (minimal)
• {1, 2, 5, 6, 10}
5
[2, 2, 3, 4, 4]
3.00
j (minimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 3, 3, 3]
2.60
k (minimal)
• {1, 2, 4, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
l (minimal)
• {1, 2, 8, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
m (minimal)
• {1, 2, 6, 9, 11}
5
[2, 2, 3, 3, 4]
2.80
n (minimal)
• {1, 2, 3, 5, 11}
5
[2, 2, 3, 3, 3]
2.60
o (minimal)
• {1, 2, 5, 10, 11}
5
[2, 2, 3, 3, 4]
2.80
p (minimal)
• {1, 2, 5, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
q (minimal)
• {1, 2, 5, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
r (minimal)
• {1, 2, 6, 10, 12}
5
[2, 2, 4, 4, 5]
3.40
s (minimal)
• {1, 2, 4, 7, 8, 10}
6
[2, 2, 3, 3, 4, 5]
3.17
t (minimal)
• {1, 2, 3, 4, 6, 10}
6
[2, 2, 3, 3, 4, 4]
3.00
u (minimal)
• {1, 2, 3, 6, 7, 8}
6
[2, 2, 3, 3, 4, 5]
3.17
v (minimal)
• {1, 2, 4, 10, 11, 12}
6
[2, 2, 3, 3, 4, 5]
3.17
w (minimal)
• {1, 2, 3, 6, 11, 12}
6
[2, 2, 3, 3, 4, 5]
3.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
3.0
5
0
18
16
2.93
6
0
5
125
3.04
7
0
0
210
3.13
8
0
0
197
3.19
9
0
0
118
3.24
10
0
0
45
3.28
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
23
722
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,8,3],[0,2,5,9],[0,9,9,1],[1,9,3,6],[1,5,8,7],[2,6,8,8],[2,7,7,6],[3,5,4,4]]
PD code (use to draw this multiloop with SnapPy): [[8,12,1,9],[9,13,10,20],[7,4,8,5],[11,3,12,4],[1,14,2,13],[10,15,11,16],[16,19,17,20],[5,17,6,18],[18,6,19,7],[2,14,3,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-17,-2)(5,2,-6,-3)(13,4,-14,-5)(6,17,-7,-18)(19,10,-20,-11)(3,12,-4,-13)(11,14,-12,-15)(15,18,-16,-19)(16,7,-9,-8)(8,9,-1,-10)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,10)(-2,5,-14,11,-20)(-3,-13,-5)(-4,13)(-6,-18,15,-12,3)(-7,16,18)(-8,-10,19,-16)(-9,8)(-11,-15,-19)(-17,6,2)(1,9,7,17)(4,12,14)
Multiloop annotated with half-edges
12^3_69 annotated with half-edges