$12^{1}_{209}$ - Minimal pinning sets
Pinning sets for 12^1_209 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_209 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 424
of which optimal: 6
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10716
on average over minimal pinning sets: 2.72667
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, 4, 5, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 3, 5, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 3, 5, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 4, 5, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
E (optimal)
• {1, 4, 5, 6, 11}
5
[2, 2, 3, 3, 3]
2.60
F (optimal)
• {1, 3, 5, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 4, 5, 8, 9}
6
[2, 2, 3, 3, 3, 4]
2.83
b (minimal)
• {1, 3, 5, 6, 10, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
c (minimal)
• {1, 3, 5, 6, 7, 11}
6
[2, 2, 3, 3, 3, 5]
3.00
d (minimal)
• {1, 4, 5, 8, 9, 12}
6
[2, 2, 3, 3, 3, 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
6
0
0
2.6
6
0
4
36
2.84
7
0
0
100
3.01
8
0
0
129
3.13
9
0
0
96
3.22
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
6
4
414
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],[0,5,5,2],[0,1,5,6],[0,7,8,4],[0,3,8,5],[1,4,2,1],[2,8,9,7],[3,6,9,9],[3,9,6,4],[6,8,7,7]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,17,9,18],[19,16,20,17],[10,5,11,6],[1,6,2,7],[18,7,19,8],[12,15,13,16],[13,4,14,5],[11,3,12,2],[3,14,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,9,-1,-10)(11,2,-12,-3)(8,3,-9,-4)(19,4,-20,-5)(16,5,-17,-6)(1,12,-2,-13)(10,13,-11,-14)(17,14,-18,-15)(6,15,-7,-16)(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,-13,10)(-2,11,13)(-3,8,18,14,-11)(-4,19,-8)(-5,16,-7,-19)(-6,-16)(-9,20,4)(-10,-14,17,5,-20)(-12,1,9,3)(-15,6,-17)(-18,7,15)(2,12)
Loop annotated with half-edges
12^1_209 annotated with half-edges