$12^{1}_{195}$ - Minimal pinning sets
Pinning sets for 12^1_195 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_195 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 254
of which optimal: 1
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05095
on average over minimal pinning sets: 2.63333
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 6, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 4, 6, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 4, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 3, 4, 6, 9, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {2, 3, 5, 6, 8, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
e (minimal)
• {2, 3, 5, 6, 9, 10}
6
[2, 2, 2, 3, 4, 4]
2.83
f (minimal)
• {2, 3, 4, 6, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
g (minimal)
• {2, 3, 4, 6, 9, 10, 12}
7
[2, 2, 2, 3, 3, 4, 5]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.4
6
0
6
7
2.64
7
0
1
47
2.87
8
0
0
79
3.04
9
0
0
69
3.15
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
7
246
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,6],[0,7,3,0],[0,2,8,8],[1,8,7,5],[1,4,9,6],[1,5,9,9],[2,9,4,8],[3,7,4,3],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[13,8,14,9],[19,10,20,11],[1,19,2,18],[3,12,4,13],[4,7,5,8],[14,5,15,6],[11,16,12,17],[2,17,3,18],[6,15,7,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(1,4,-2,-5)(10,5,-11,-6)(16,7,-17,-8)(20,9,-1,-10)(3,12,-4,-13)(8,13,-9,-14)(19,14,-20,-15)(15,18,-16,-19)(6,17,-7,-18)
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,-5,10)(-2,11,5)(-3,-13,8,-17,6,-11)(-4,1,9,13)(-6,-18,15,-20,-10)(-7,16,18)(-8,-14,19,-16)(-9,20,14)(-12,3)(-15,-19)(2,4,12)(7,17)
Loop annotated with half-edges
12^1_195 annotated with half-edges