$12^{2}_{287}$ - Minimal pinning sets
Pinning sets for 12^2_287 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_287 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 416
of which optimal: 7
of which minimal: 13
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.17366
on average over minimal pinning sets: 2.8978
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)
• {2, 4, 6, 9, 11}
5
[2, 3, 3, 3, 3]
2.80
B (optimal)
• {2, 4, 6, 8, 11}
5
[2, 3, 3, 3, 3]
2.80
C (optimal)
• {2, 4, 6, 10, 11}
5
[2, 3, 3, 3, 6]
3.40
D (optimal)
• {2, 4, 5, 8, 11}
5
[2, 3, 3, 3, 3]
2.80
E (optimal)
• {2, 4, 5, 6, 11}
5
[2, 3, 3, 3, 3]
2.80
F (optimal)
• {2, 3, 4, 6, 11}
5
[2, 3, 3, 3, 3]
2.80
G (optimal)
• {2, 4, 6, 7, 11}
5
[2, 3, 3, 3, 5]
3.20
a (minimal)
• {1, 3, 4, 6, 11, 12}
6
[2, 3, 3, 3, 3, 3]
2.83
b (minimal)
• {1, 3, 4, 6, 9, 12}
6
[2, 3, 3, 3, 3, 3]
2.83
c (minimal)
• {1, 2, 4, 6, 9, 12}
6
[2, 3, 3, 3, 3, 3]
2.83
d (minimal)
• {1, 3, 4, 5, 8, 11, 12}
7
[2, 3, 3, 3, 3, 3, 3]
2.86
e (minimal)
• {1, 3, 4, 5, 8, 9, 12}
7
[2, 3, 3, 3, 3, 3, 3]
2.86
f (minimal)
• {1, 2, 4, 5, 8, 9, 12}
7
[2, 3, 3, 3, 3, 3, 3]
2.86
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
3
33
3.02
7
0
3
85
3.09
8
0
0
124
3.17
9
0
0
102
3.24
10
0
0
47
3.29
11
0
0
11
3.32
12
0
0
1
3.33
Total
7
6
403
Other information about this multiloop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,7],[0,8,5,1],[1,4,8,2],[3,8,9,7],[3,6,9,9],[4,9,6,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,13,6,12],[3,11,4,12],[19,10,20,11],[1,14,2,13],[6,2,7,3],[15,18,16,19],[16,9,17,10],[14,8,15,7],[8,17,9,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(15,8,-16,-9)(18,9,-19,-10)(1,10,-2,-11)(11,20,-12,-5)(12,3,-13,-4)(7,16,-8,-17)(14,17,-15,-18)(2,19,-3,-20)(6,13,-7,-14)
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,-11,-5)(-2,-20,11)(-3,12,20)(-4,5,-12)(-6,-14,-18,-10,1)(-7,-17,14)(-8,15,17)(-9,18,-15)(-13,6,4)(-16,7,13,3,19,9)(-19,2,10)(8,16)
Multiloop annotated with half-edges
12^2_287 annotated with half-edges