$12^{1}_{105}$ - Minimal pinning sets
Pinning sets for 12^1_105 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_105 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 496
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: 3.1207
on average over minimal pinning sets: 2.75
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)
• {1, 3, 8, 11}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 6, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 2, 5, 8, 11}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {1, 3, 6, 11, 12}
5
[2, 2, 3, 3, 3]
2.60
d (minimal)
• {1, 3, 6, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
e (minimal)
• {1, 2, 6, 9, 11, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
f (minimal)
• {1, 2, 6, 8, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
g (minimal)
• {1, 2, 5, 6, 11, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
h (minimal)
• {1, 2, 4, 8, 9, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
i (minimal)
• {1, 2, 4, 7, 8, 11}
6
[2, 2, 3, 3, 4, 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
2.5
5
0
4
8
2.75
6
0
5
51
2.92
7
0
0
123
3.05
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
486
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 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],[0,2,6,7],[0,5,1,1],[1,4,8,2],[2,8,9,3],[3,9,9,8],[5,7,9,6],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[3,20,4,1],[2,11,3,12],[14,19,15,20],[4,15,5,16],[1,13,2,12],[13,10,14,11],[18,5,19,6],[16,8,17,7],[9,6,10,7],[17,8,18,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,3,-17,-4)(7,4,-8,-5)(5,14,-6,-15)(15,6,-16,-7)(11,8,-12,-9)(20,9,-1,-10)(10,19,-11,-20)(2,13,-3,-14)(12,17,-13,-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,-19,10)(-2,-14,5,-8,11,19)(-3,16,6,14)(-4,7,-16)(-5,-15,-7)(-6,15)(-9,20,-11)(-10,-20)(-12,-18,1,9)(-13,2,18)(-17,12,8,4)(3,13,17)
Loop annotated with half-edges
12^1_105 annotated with half-edges