$12^{1}_{109}$ - Minimal pinning sets
Pinning sets for 12^1_109 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_109 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 420
of which optimal: 6
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11545
on average over minimal pinning sets: 2.7037
on average over optimal pinning sets: 2.66667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 9, 10, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {2, 3, 8, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {2, 3, 6, 9, 10}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {1, 2, 8, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
E (optimal)
• {1, 2, 6, 9, 10}
5
[2, 2, 3, 3, 4]
2.80
F (optimal)
• {2, 3, 7, 9, 10}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 4, 9, 10, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
b (minimal)
• {1, 2, 7, 8, 9, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 2, 4, 7, 9, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
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
2.67
6
0
3
36
2.87
7
0
0
98
3.02
8
0
0
128
3.14
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
3
411
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],[0,4,5,6],[0,6,7,3],[0,2,8,9],[0,5,5,1],[1,4,4,6],[1,5,9,2],[2,9,8,8],[3,7,7,9],[3,8,7,6]]
PD code (use to draw this loop with SnapPy): [[15,20,16,1],[14,11,15,12],[4,19,5,20],[16,5,17,6],[1,13,2,12],[2,13,3,14],[3,10,4,11],[7,18,8,19],[17,8,18,9],[6,9,7,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,2,-14,-3)(14,5,-15,-6)(3,6,-4,-7)(20,7,-1,-8)(11,8,-12,-9)(9,18,-10,-19)(19,10,-20,-11)(4,15,-5,-16)(1,16,-2,-17)(12,17,-13,-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,-17,12,8)(-2,13,17)(-3,-7,20,10,18,-13)(-4,-16,1,7)(-5,14,2,16)(-6,3,-14)(-8,11,-20)(-9,-19,-11)(-10,19)(-12,-18,9)(-15,4,6)(5,15)
Loop annotated with half-edges
12^1_109 annotated with half-edges