$12^{3}_{74}$ - Minimal pinning sets
Pinning sets for 12^3_74 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_74 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 767
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: 3.16223
on average over minimal pinning sets: 2.91467
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, 3, 6, 10}
4
[2, 2, 4, 4]
3.00
B (optimal)
• {1, 3, 5, 9}
4
[2, 2, 4, 4]
3.00
a (minimal)
• {1, 3, 4, 11, 12}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 3, 4, 10, 12}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {1, 3, 4, 5, 12}
5
[2, 2, 3, 3, 4]
2.80
d (minimal)
• {1, 3, 5, 10, 12}
5
[2, 2, 3, 4, 4]
3.00
e (minimal)
• {1, 3, 4, 7, 12}
5
[2, 2, 3, 3, 3]
2.60
f (minimal)
• {1, 3, 4, 6, 11}
5
[2, 2, 3, 4, 4]
3.00
g (minimal)
• {1, 3, 4, 6, 7}
5
[2, 2, 3, 3, 4]
2.80
h (minimal)
• {1, 2, 3, 10, 12}
5
[2, 2, 3, 3, 4]
2.80
i (minimal)
• {1, 2, 3, 5, 12}
5
[2, 2, 3, 3, 4]
2.80
j (minimal)
• {1, 2, 3, 7, 12}
5
[2, 2, 3, 3, 3]
2.60
k (minimal)
• {1, 2, 3, 6, 7}
5
[2, 2, 3, 3, 4]
2.80
l (minimal)
• {1, 3, 4, 9, 11}
5
[2, 2, 3, 4, 4]
3.00
m (minimal)
• {1, 3, 4, 7, 9}
5
[2, 2, 3, 3, 4]
2.80
n (minimal)
• {1, 3, 6, 7, 9}
5
[2, 2, 3, 4, 4]
3.00
o (minimal)
• {1, 2, 3, 7, 9}
5
[2, 2, 3, 3, 4]
2.80
p (minimal)
• {1, 2, 3, 8, 10}
5
[2, 2, 3, 4, 4]
3.00
q (minimal)
• {1, 2, 3, 5, 8}
5
[2, 2, 3, 4, 4]
3.00
r (minimal)
• {1, 2, 3, 7, 8}
5
[2, 2, 3, 3, 4]
2.80
s (minimal)
• {1, 2, 3, 5, 6, 11}
6
[2, 2, 3, 4, 4, 4]
3.17
t (minimal)
• {1, 2, 3, 9, 10, 11}
6
[2, 2, 3, 4, 4, 4]
3.17
u (minimal)
• {1, 3, 4, 5, 6, 8}
6
[2, 2, 3, 4, 4, 4]
3.17
v (minimal)
• {1, 2, 3, 4, 8, 11}
6
[2, 2, 3, 3, 4, 4]
3.00
w (minimal)
• {1, 3, 4, 8, 9, 10}
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
4
2
0
0
3.0
5
0
18
16
2.96
6
0
5
128
3.07
7
0
0
218
3.14
8
0
0
204
3.2
9
0
0
120
3.24
10
0
0
45
3.28
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
23
742
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: [[1,2,2,3],[0,4,5,6],[0,6,3,0],[0,2,7,4],[1,3,7,8],[1,8,9,6],[1,5,7,2],[3,6,9,4],[4,9,9,5],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,17,10,20],[15,7,16,8],[1,7,2,6],[17,6,18,5],[10,13,11,14],[14,19,15,20],[2,19,3,18],[12,4,13,5],[11,4,12,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,9,-1,-10)(10,1,-11,-2)(2,17,-3,-18)(3,6,-4,-7)(13,4,-14,-5)(16,11,-9,-12)(5,14,-6,-15)(18,7,-19,-8)(12,19,-13,-20)(20,15,-17,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,-18,-8,-10)(-3,-7,18)(-4,13,19,7)(-5,-15,20,-13)(-6,3,17,15)(-9,8,-19,12)(-11,16,-17,2)(-12,-20,-16)(-14,5)(1,9,11)(4,6,14)
Multiloop annotated with half-edges
12^3_74 annotated with half-edges