$12^{1}_{288}$ - Minimal pinning sets
Pinning sets for 12^1_288 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_288 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 552
of which optimal: 2
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12195
on average over minimal pinning sets: 2.775
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {3, 4, 5, 11}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {3, 5, 7, 11}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {2, 4, 5, 10, 11}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {3, 5, 6, 8, 11}
5
[2, 2, 3, 4, 5]
3.20
c (minimal)
• {2, 4, 5, 9, 11}
5
[2, 2, 3, 3, 4]
2.80
d (minimal)
• {3, 5, 8, 9, 11}
5
[2, 2, 3, 4, 4]
3.00
e (minimal)
• {1, 2, 4, 5, 11}
5
[2, 2, 3, 3, 3]
2.60
f (minimal)
• {1, 3, 5, 8, 11}
5
[2, 2, 3, 3, 4]
2.80
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
2.5
5
0
6
15
2.79
6
0
0
76
2.96
7
0
0
141
3.08
8
0
0
154
3.16
9
0
0
104
3.23
10
0
0
43
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
6
544
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,7,3],[0,2,8,9],[0,9,9,5],[1,4,8,6],[1,5,8,7],[1,6,2,2],[3,6,5,9],[3,8,4,4]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[17,12,18,13],[19,4,20,5],[14,4,15,3],[1,9,2,8],[16,7,17,8],[11,6,12,7],[18,6,19,5],[15,11,16,10],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,12,-2,-13)(17,2,-18,-3)(3,16,-4,-17)(9,4,-10,-5)(14,5,-15,-6)(6,19,-7,-20)(20,7,-1,-8)(8,13,-9,-14)(15,10,-16,-11)(11,18,-12,-19)
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,-13,8)(-2,17,-4,9,13)(-3,-17)(-5,14,-9)(-6,-20,-8,-14)(-7,20)(-10,15,5)(-11,-19,6,-15)(-12,1,7,19)(-16,3,-18,11)(2,12,18)(4,16,10)
Loop annotated with half-edges
12^1_288 annotated with half-edges