$12^{3}_{75}$ - Minimal pinning sets
Pinning sets for 12^3_75 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_75 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 612
of which optimal: 18
of which minimal: 18
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.1454
on average over minimal pinning sets: 2.84444
on average over optimal pinning sets: 2.84444
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 10}
5
[2, 2, 3, 3, 4]
2.80
B (optimal)
• {2, 3, 4, 5, 8}
5
[2, 2, 3, 3, 4]
2.80
C (optimal)
• {2, 3, 5, 8, 12}
5
[2, 2, 3, 4, 4]
3.00
D (optimal)
• {1, 2, 3, 5, 8}
5
[2, 2, 3, 3, 4]
2.80
E (optimal)
• {2, 3, 4, 5, 11}
5
[2, 2, 3, 3, 4]
2.80
F (optimal)
• {2, 3, 5, 7, 11}
5
[2, 2, 3, 4, 4]
3.00
G (optimal)
• {1, 2, 3, 5, 11}
5
[2, 2, 3, 3, 4]
2.80
H (optimal)
• {2, 3, 5, 8, 11}
5
[2, 2, 3, 4, 4]
3.00
I (optimal)
• {2, 3, 4, 6, 9}
5
[2, 2, 3, 3, 4]
2.80
J (optimal)
• {2, 3, 4, 6, 12}
5
[2, 2, 3, 3, 4]
2.80
K (optimal)
• {2, 3, 4, 6, 7}
5
[2, 2, 3, 3, 4]
2.80
L (optimal)
• {2, 3, 6, 7, 12}
5
[2, 2, 3, 4, 4]
3.00
M (optimal)
• {2, 3, 4, 5, 6}
5
[2, 2, 3, 3, 3]
2.60
N (optimal)
• {1, 2, 3, 6, 12}
5
[2, 2, 3, 3, 4]
2.80
O (optimal)
• {1, 2, 3, 6, 7}
5
[2, 2, 3, 3, 4]
2.80
P (optimal)
• {1, 2, 3, 5, 6}
5
[2, 2, 3, 3, 3]
2.60
Q (optimal)
• {2, 3, 6, 8, 12}
5
[2, 2, 3, 4, 4]
3.00
R (optimal)
• {2, 3, 6, 7, 11}
5
[2, 2, 3, 4, 4]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
18
0
0
2.84
6
0
0
85
3.0
7
0
0
166
3.11
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
18
0
594
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],[0,1,6,7],[0,7,4,4],[0,3,3,5],[1,4,7,8],[1,8,9,2],[2,9,5,3],[5,9,9,6],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,3,1,4],[4,11,5,14],[9,13,10,14],[2,20,3,15],[1,20,2,19],[11,19,12,18],[5,8,6,9],[12,15,13,16],[7,17,8,18],[6,17,7,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(3,6,-4,-7)(7,20,-8,-17)(17,2,-18,-3)(13,4,-14,-5)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,20,12)(-2,17,-8)(-3,-7,-17)(-4,13,-20,7)(-5,-15,-19,-13)(-6,3,-18,15)(-9,16,18,2)(-10,-12,19,-16)(-11,10)(-14,5)(1,11,9)(4,6,14)
Multiloop annotated with half-edges
12^3_75 annotated with half-edges