$12^{1}_{60}$ - Minimal pinning sets
Pinning sets for 12^1_60 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_60 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 286
of which optimal: 11
of which minimal: 12
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10959
on average over minimal pinning sets: 2.7381
on average over optimal pinning sets: 2.72727
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 8, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
B (optimal)
• {1, 2, 3, 6, 7, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
C (optimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 3, 3, 3, 4]
2.83
D (optimal)
• {1, 2, 3, 5, 6, 8}
6
[2, 2, 3, 3, 3, 3]
2.67
E (optimal)
• {1, 4, 5, 6, 7, 8}
6
[2, 2, 3, 3, 3, 3]
2.67
F (optimal)
• {1, 4, 5, 6, 7, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
G (optimal)
• {1, 4, 5, 6, 8, 9}
6
[2, 2, 3, 3, 3, 4]
2.83
H (optimal)
• {1, 3, 4, 6, 7, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
I (optimal)
• {1, 3, 4, 6, 8, 9}
6
[2, 2, 3, 3, 3, 4]
2.83
J (optimal)
• {1, 4, 5, 6, 8, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
K (optimal)
• {1, 2, 4, 5, 6, 8}
6
[2, 2, 3, 3, 3, 3]
2.67
a (minimal)
• {1, 3, 4, 6, 8, 10, 11}
7
[2, 2, 3, 3, 3, 3, 4]
2.86
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
11
0
0
2.73
7
0
1
50
2.93
8
0
0
91
3.09
9
0
0
82
3.19
10
0
0
40
3.26
11
0
0
10
3.31
12
0
0
1
3.33
Total
11
1
274
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,3,0],[0,2,6,1],[1,6,7,5],[1,4,7,8],[2,9,4,3],[4,9,8,5],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,9,1,10],[10,18,11,17],[8,19,9,20],[1,19,2,18],[11,7,12,6],[16,5,17,6],[7,2,8,3],[12,15,13,16],[13,4,14,5],[3,14,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(13,2,-14,-3)(3,8,-4,-9)(17,6,-18,-7)(20,11,-1,-12)(9,12,-10,-13)(14,7,-15,-8)(15,18,-16,-19)(5,16,-6,-17)(4,19,-5,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,12)(-2,13,-10)(-3,-9,-13)(-4,-20,-12,9)(-5,-17,-7,14,2,-11,20)(-6,17)(-8,3,-14)(-15,-19,4,8)(-16,5,19)(-18,15,7)(1,11)(6,16,18)
Loop annotated with half-edges
12^1_60 annotated with half-edges