$12^{1}_{345}$ - Minimal pinning sets
Pinning sets for 12^1_345 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_345 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 284
of which optimal: 3
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04615
on average over minimal pinning sets: 2.54667
on average over optimal pinning sets: 2.46667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 5, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 2, 3, 4, 5, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 4, 5, 8, 11}
6
[2, 2, 2, 3, 3, 5]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.47
6
0
2
20
2.71
7
0
0
60
2.91
8
0
0
85
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
3
2
279
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: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,6,6,7],[0,8,9,1],[1,9,9,2],[2,7,3,3],[3,6,8,8],[4,7,7,9],[4,8,5,5]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[4,9,5,10],[19,8,20,9],[6,14,7,13],[1,11,2,10],[18,3,19,4],[7,14,8,15],[15,12,16,13],[11,16,12,17],[2,17,3,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(16,5,-17,-6)(1,6,-2,-7)(7,18,-8,-19)(12,9,-13,-10)(10,3,-11,-4)(4,11,-5,-12)(8,13,-9,-14)(19,14,-20,-15)(2,17,-3,-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,-7,-19,-15)(-2,-18,7)(-3,10,-13,8,18)(-4,-12,-10)(-5,16,20,14,-9,12)(-6,1,-16)(-8,-14,19)(-11,4)(-17,2,6)(-20,15)(3,17,5,11)(9,13)
Loop annotated with half-edges
12^1_345 annotated with half-edges