$12^{1}_{287}$ - Minimal pinning sets
Pinning sets for 12^1_287 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_287 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 708
of which optimal: 3
of which minimal: 15
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.1537
on average over minimal pinning sets: 2.86
on average over optimal pinning sets: 2.83333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 5, 8, 9}
4
[2, 2, 4, 4]
3.00
B (optimal)
• {1, 4, 5, 9}
4
[2, 2, 3, 4]
2.75
C (optimal)
• {1, 5, 9, 11}
4
[2, 2, 3, 4]
2.75
a (minimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 4, 6, 9, 10}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {1, 2, 5, 9, 10}
5
[2, 2, 4, 4, 4]
3.20
d (minimal)
• {1, 2, 5, 7, 9}
5
[2, 2, 4, 4, 4]
3.20
e (minimal)
• {1, 3, 4, 9, 12}
5
[2, 2, 3, 3, 4]
2.80
f (minimal)
• {1, 5, 9, 10, 12}
5
[2, 2, 4, 4, 4]
3.20
g (minimal)
• {1, 5, 7, 9, 12}
5
[2, 2, 4, 4, 4]
3.20
h (minimal)
• {1, 3, 7, 9, 11}
5
[2, 2, 3, 3, 4]
2.80
i (minimal)
• {1, 3, 4, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
j (minimal)
• {1, 3, 6, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
k (minimal)
• {1, 2, 6, 9, 11}
5
[2, 2, 3, 3, 4]
2.80
l (minimal)
• {1, 4, 6, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.83
5
0
12
21
2.95
6
0
0
113
3.04
7
0
0
193
3.13
8
0
0
192
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
3
12
693
Other information about this loop 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],[0,5,6,6],[0,6,7,3],[0,2,8,9],[0,9,9,5],[1,4,8,7],[1,7,2,1],[2,6,5,8],[3,7,5,9],[3,8,4,4]]
PD code (use to draw this loop with SnapPy): [[7,20,8,1],[11,6,12,7],[19,4,20,5],[8,4,9,3],[1,16,2,17],[17,10,18,11],[5,12,6,13],[13,18,14,19],[9,14,10,15],[15,2,16,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,5,-1,-6)(16,1,-17,-2)(8,3,-9,-4)(14,7,-15,-8)(2,9,-3,-10)(6,11,-7,-12)(18,13,-19,-14)(10,15,-11,-16)(4,17,-5,-18)(12,19,-13,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-11,6)(-2,-10,-16)(-3,8,-15,10)(-4,-18,-14,-8)(-5,20,-13,18)(-6,-12,-20)(-7,14,-19,12)(-9,2,-17,4)(1,5,17)(3,9)(7,11,15)(13,19)
Loop annotated with half-edges
12^1_287 annotated with half-edges