$12^{1}_{118}$ - Minimal pinning sets
Pinning sets for 12^1_118 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_118 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 600
of which optimal: 2
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.1293
on average over minimal pinning sets: 2.71667
on average over optimal pinning sets: 2.625
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 7, 11}
4
[2, 2, 3, 4]
2.75
B (optimal)
• {1, 3, 8, 11}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 4, 9, 11, 12}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 4, 8, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 4, 5, 11, 12}
5
[2, 2, 3, 3, 4]
2.80
d (minimal)
• {1, 4, 5, 8, 11}
5
[2, 2, 3, 3, 4]
2.80
e (minimal)
• {1, 3, 6, 11, 12}
5
[2, 2, 3, 3, 4]
2.80
f (minimal)
• {1, 3, 6, 7, 11}
5
[2, 2, 3, 4, 4]
3.00
g (minimal)
• {1, 3, 4, 11, 12}
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
2
0
0
2.62
5
0
7
16
2.81
6
0
0
85
2.97
7
0
0
157
3.09
8
0
0
168
3.17
9
0
0
110
3.23
10
0
0
44
3.28
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
7
591
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],[0,4,4,5],[0,5,6,3],[0,2,7,8],[0,5,1,1],[1,4,8,2],[2,8,9,7],[3,6,9,9],[3,9,6,5],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,9,3,10],[12,19,13,20],[4,13,5,14],[1,11,2,10],[11,8,12,9],[15,18,16,19],[5,16,6,17],[14,7,15,8],[17,6,18,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,4,-16,-5)(12,5,-13,-6)(9,6,-10,-7)(20,7,-1,-8)(8,19,-9,-20)(2,11,-3,-12)(3,14,-4,-15)(13,16,-14,-17)(10,17,-11,-18)(1,18,-2,-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,-19,8)(-2,-12,-6,9,19)(-3,-15,-5,12)(-4,15)(-7,20,-9)(-8,-20)(-10,-18,1,7)(-11,2,18)(-13,-17,10,6)(-14,3,11,17)(-16,13,5)(4,14,16)
Loop annotated with half-edges
12^1_118 annotated with half-edges