$12^{1}_{182}$ - Minimal pinning sets
Pinning sets for 12^1_182 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_182 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 312
of which optimal: 4
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04935
on average over minimal pinning sets: 2.55
on average over optimal pinning sets: 2.45
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 11, 12}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 2, 3, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 8, 11, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
b (minimal)
• {1, 2, 4, 8, 9, 11}
6
[2, 2, 2, 3, 4, 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
4
0
0
2.45
6
0
2
24
2.72
7
0
0
68
2.92
8
0
0
94
3.07
9
0
0
75
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
4
2
306
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 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],[0,7,7,8],[0,9,6,5],[1,4,2,1],[2,4,9,7],[3,6,8,3],[3,7,9,9],[4,8,8,6]]
PD code (use to draw this loop with SnapPy): [[20,5,1,6],[6,18,7,17],[19,16,20,17],[4,11,5,12],[1,9,2,8],[18,8,19,7],[2,15,3,16],[12,3,13,4],[13,10,14,11],[9,14,10,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(3,14,-4,-15)(4,17,-5,-18)(9,6,-10,-7)(20,7,-1,-8)(5,10,-6,-11)(18,11,-19,-12)(15,12,-16,-13)(13,2,-14,-3)(8,19,-9,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,12,-19,8)(-2,13,-16)(-3,-15,-13)(-4,-18,-12,15)(-5,-11,18)(-6,9,19,11)(-7,20,-9)(-8,-20)(-10,5,17,1,7)(-14,3)(-17,4,14,2)(6,10)
Loop annotated with half-edges
12^1_182 annotated with half-edges