$12^{1}_{283}$ - Minimal pinning sets
Pinning sets for 12^1_283 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_283 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 336
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: 3.11035
on average over minimal pinning sets: 2.72593
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 4, 5, 9, 12}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 3, 5, 8, 9, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
b (minimal)
• {1, 3, 4, 5, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 2, 4, 5, 7, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
d (minimal)
• {1, 2, 4, 5, 9, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
e (minimal)
• {1, 3, 5, 7, 9, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
f (minimal)
• {1, 3, 5, 8, 9, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
g (minimal)
• {1, 4, 5, 7, 11, 12}
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
5
2
0
0
2.6
6
0
7
14
2.79
7
0
0
68
2.97
8
0
0
105
3.11
9
0
0
88
3.2
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
7
327
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,7,3],[0,2,8,4],[0,3,8,9],[1,9,9,6],[1,5,8,7],[1,6,2,2],[3,6,9,4],[4,8,5,5]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[17,4,18,5],[12,19,13,20],[6,13,7,14],[1,14,2,15],[9,16,10,17],[10,3,11,4],[18,11,19,12],[7,3,8,2],[15,8,16,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,5,-1,-6)(8,1,-9,-2)(15,2,-16,-3)(13,6,-14,-7)(16,9,-17,-10)(4,11,-5,-12)(19,12,-20,-13)(7,14,-8,-15)(10,17,-11,-18)(3,18,-4,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,14,6)(-2,15,-8)(-3,-19,-13,-7,-15)(-4,-12,19)(-5,20,12)(-6,13,-20)(-9,16,2)(-10,-18,3,-16)(-11,4,18)(-14,7)(-17,10)(1,5,11,17,9)
Loop annotated with half-edges
12^1_283 annotated with half-edges