$12^{1}_{284}$ - Minimal pinning sets
Pinning sets for 12^1_284 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_284 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 360
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: 3.11258
on average over minimal pinning sets: 2.65556
on average over optimal pinning sets: 2.65
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 5, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {2, 3, 5, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {2, 4, 6, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {2, 4, 5, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 4, 7, 8, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 2, 3, 4, 7, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
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
2.65
6
0
2
26
2.85
7
0
0
77
3.0
8
0
0
110
3.12
9
0
0
89
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
4
2
354
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: [[1,2,2,3],[0,4,5,6],[0,7,7,0],[0,8,9,4],[1,3,9,5],[1,4,9,6],[1,5,8,7],[2,6,8,2],[3,7,6,9],[3,8,5,4]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[11,4,12,5],[19,6,20,7],[1,14,2,15],[15,10,16,11],[16,3,17,4],[12,17,13,18],[7,18,8,19],[8,13,9,14],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,20,-14,-1)(16,3,-17,-4)(9,4,-10,-5)(2,7,-3,-8)(15,8,-16,-9)(6,11,-7,-12)(1,12,-2,-13)(19,14,-20,-15)(10,17,-11,-18)(5,18,-6,-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)(-2,-8,15,-20,13)(-3,16,8)(-4,9,-16)(-5,-19,-15,-9)(-6,-12,1,-14,19)(-7,2,12)(-10,-18,5)(-11,6,18)(-17,10,4)(3,7,11,17)(14,20)
Loop annotated with half-edges
12^1_284 annotated with half-edges