$12^{1}_{58}$ - Minimal pinning sets
Pinning sets for 12^1_58 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_58 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 276
of which optimal: 2
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04954
on average over minimal pinning sets: 2.5625
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {2, 4, 7, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 2, 3, 6, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {2, 3, 6, 7, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {2, 4, 6, 9, 10, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {2, 3, 6, 9, 10, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
e (minimal)
• {2, 4, 6, 8, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
f (minimal)
• {2, 3, 6, 8, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
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
2.5
6
0
6
13
2.69
7
0
0
57
2.9
8
0
0
84
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
6
268
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,3],[0,2,7,8],[0,9,9,1],[1,9,9,6],[1,5,8,2],[2,8,8,3],[3,7,7,6],[4,5,5,4]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[15,4,16,5],[19,10,20,11],[6,10,7,9],[1,14,2,15],[3,12,4,13],[16,12,17,11],[18,7,19,8],[8,17,9,18],[13,2,14,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(15,4,-16,-5)(6,3,-7,-4)(16,7,-17,-8)(8,13,-9,-14)(20,9,-1,-10)(18,11,-19,-12)(5,14,-6,-15)(2,17,-3,-18)(10,19,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-19,10)(-2,-18,-12)(-3,6,14,-9,20,-11,18)(-4,15,-6)(-5,-15)(-7,16,4)(-8,-14,5,-16)(-10,-20)(-13,8,-17,2)(1,9,13)(3,17,7)(11,19)
Loop annotated with half-edges
12^1_58 annotated with half-edges