$12^{1}_{260}$ - Minimal pinning sets
Pinning sets for 12^1_260 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_260 Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 188
of which optimal: 5
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04154
on average over minimal pinning sets: 2.56463
on average over optimal pinning sets: 2.53333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
B (optimal)
• {1, 3, 5, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
C (optimal)
• {1, 3, 5, 7, 10, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
D (optimal)
• {1, 2, 3, 5, 7, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
E (optimal)
• {1, 3, 4, 5, 7, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
a (minimal)
• {1, 2, 3, 5, 6, 9, 10}
7
[2, 2, 2, 3, 3, 3, 4]
2.71
b (minimal)
• {1, 3, 4, 5, 6, 9, 10}
7
[2, 2, 2, 3, 3, 3, 3]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
5
0
0
2.53
7
0
2
25
2.78
8
0
0
56
2.98
9
0
0
58
3.12
10
0
0
32
3.22
11
0
0
9
3.29
12
0
0
1
3.33
Total
5
2
181
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,6,7,7],[1,8,9,5],[2,4,9,2],[3,9,8,7],[3,6,8,3],[4,7,6,9],[4,8,6,5]]
PD code (use to draw this loop with SnapPy): [[20,7,1,8],[8,19,9,20],[9,6,10,7],[1,14,2,15],[18,11,19,12],[5,10,6,11],[13,16,14,17],[2,16,3,15],[12,3,13,4],[4,17,5,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(15,2,-16,-3)(20,5,-1,-6)(13,6,-14,-7)(16,9,-17,-10)(10,17,-11,-18)(8,11,-9,-12)(19,12,-20,-13)(3,14,-4,-15)(7,18,-8,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,14,6)(-2,15,-4)(-3,-15)(-5,20,12,-9,16,2)(-6,13,-20)(-7,-19,-13)(-8,-12,19)(-10,-18,7,-14,3,-16)(-11,8,18)(-17,10)(1,5)(9,11,17)
Loop annotated with half-edges
12^1_260 annotated with half-edges