$12^{1}_{281}$ - Minimal pinning sets
Pinning sets for 12^1_281 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_281 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 612
of which optimal: 1
of which minimal: 12
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.18729
on average over minimal pinning sets: 2.91528
on average over optimal pinning sets: 2.75
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 10}
4
[2, 3, 3, 3]
2.75
a (minimal)
• {1, 2, 4, 10, 11}
5
[2, 3, 3, 3, 3]
2.80
b (minimal)
• {1, 4, 7, 10, 11}
5
[2, 3, 3, 3, 4]
3.00
c (minimal)
• {1, 4, 8, 9, 11}
5
[2, 3, 3, 3, 4]
3.00
d (minimal)
• {1, 4, 8, 10, 11}
5
[2, 3, 3, 3, 4]
3.00
e (minimal)
• {1, 2, 3, 5, 10}
5
[2, 3, 3, 3, 3]
2.80
f (minimal)
• {1, 3, 5, 9, 10}
5
[2, 3, 3, 3, 3]
2.80
g (minimal)
• {1, 3, 5, 7, 10}
5
[2, 3, 3, 3, 4]
3.00
h (minimal)
• {1, 3, 5, 10, 12}
5
[2, 3, 3, 3, 5]
3.20
i (minimal)
• {1, 3, 5, 9, 11}
5
[2, 3, 3, 3, 3]
2.80
j (minimal)
• {1, 2, 4, 5, 9, 11}
6
[2, 3, 3, 3, 3, 3]
2.83
k (minimal)
• {1, 4, 5, 7, 9, 11}
6
[2, 3, 3, 3, 3, 4]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.75
5
0
9
8
2.93
6
0
2
73
3.05
7
0
0
155
3.14
8
0
0
179
3.21
9
0
0
123
3.26
10
0
0
50
3.3
11
0
0
11
3.32
12
0
0
1
3.33
Total
1
11
600
Other information about this loop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,7],[0,7,7,8],[0,8,9,5],[1,4,9,6],[1,5,9,2],[2,8,3,3],[3,7,9,4],[4,8,6,5]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[4,9,5,10],[19,8,20,9],[6,14,7,13],[1,16,2,17],[10,17,11,18],[18,3,19,4],[7,14,8,15],[15,12,16,13],[2,12,3,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(16,5,-17,-6)(1,6,-2,-7)(8,19,-9,-20)(12,9,-13,-10)(10,3,-11,-4)(4,11,-5,-12)(18,13,-19,-14)(7,14,-8,-15)(2,17,-3,-18)
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,-7,-15)(-2,-18,-14,7)(-3,10,-13,18)(-4,-12,-10)(-5,16,20,-9,12)(-6,1,-16)(-8,-20,15)(-11,4)(-17,2,6)(-19,8,14)(3,17,5,11)(9,19,13)
Loop annotated with half-edges
12^1_281 annotated with half-edges