$12^{1}_{54}$ - Minimal pinning sets
Pinning sets for 12^1_54 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_54 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 500
of which optimal: 1
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12612
on average over minimal pinning sets: 2.66167
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, 7, 11}
4
[2, 2, 3, 4]
2.75
a (minimal)
• {1, 3, 9, 11, 12}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 3, 8, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 3, 5, 11, 12}
5
[2, 2, 3, 3, 3]
2.60
d (minimal)
• {1, 3, 5, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
e (minimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
f (minimal)
• {1, 2, 4, 9, 11, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
g (minimal)
• {1, 2, 4, 8, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
h (minimal)
• {1, 2, 4, 5, 11, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
i (minimal)
• {1, 2, 4, 5, 8, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
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
5
8
2.82
6
0
4
54
2.94
7
0
0
124
3.06
8
0
0
147
3.16
9
0
0
103
3.23
10
0
0
43
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
9
490
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 5, 6]
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],[1,4,6,2],[2,5,8,9],[3,9,9,8],[3,7,9,6],[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],[18,7,19,8],[5,16,6,17],[14,17,15,18],[15,6,16,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,4,-14,-5)(16,5,-17,-6)(9,6,-10,-7)(20,7,-1,-8)(8,19,-9,-20)(2,11,-3,-12)(3,14,-4,-15)(12,15,-13,-16)(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,-16,-6,9,19)(-3,-15,12)(-4,13,15)(-5,16,-13)(-7,20,-9)(-8,-20)(-10,-18,1,7)(-11,2,18)(-14,3,11,17,5)(-17,10,6)(4,14)
Loop annotated with half-edges
12^1_54 annotated with half-edges