$11^{2}_{50}$ - Minimal pinning sets
Pinning sets for 11^2_50 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_50 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 244
of which optimal: 7
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12568
on average over minimal pinning sets: 2.86667
on average over optimal pinning sets: 2.82857
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 6, 10, 11}
5
[2, 3, 3, 3, 3]
2.80
B (optimal)
• {1, 2, 6, 7, 11}
5
[2, 3, 3, 3, 3]
2.80
C (optimal)
• {1, 2, 7, 8, 10}
5
[2, 3, 3, 3, 3]
2.80
D (optimal)
• {2, 3, 7, 8, 10}
5
[2, 3, 3, 3, 4]
3.00
E (optimal)
• {2, 4, 7, 8, 10}
5
[2, 3, 3, 3, 3]
2.80
F (optimal)
• {2, 4, 6, 8, 10}
5
[2, 3, 3, 3, 3]
2.80
G (optimal)
• {1, 2, 7, 8, 11}
5
[2, 3, 3, 3, 3]
2.80
a (minimal)
• {1, 2, 4, 6, 9, 11}
6
[2, 3, 3, 3, 3, 4]
3.00
b (minimal)
• {2, 3, 6, 7, 10, 11}
6
[2, 3, 3, 3, 3, 4]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
7
0
0
2.83
6
0
2
35
2.98
7
0
0
75
3.1
8
0
0
75
3.18
9
0
0
39
3.23
10
0
0
10
3.26
11
0
0
1
3.27
Total
7
2
235
Other information about this multiloop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,3],[0,2,7,7],[0,7,8,5],[1,4,8,6],[1,5,8,2],[3,8,4,3],[4,7,6,5]]
PD code (use to draw this multiloop with SnapPy): [[5,12,6,1],[4,18,5,13],[11,17,12,18],[6,17,7,16],[1,8,2,9],[13,9,14,10],[10,3,11,4],[7,15,8,16],[2,15,3,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,12,-8,-1)(10,5,-11,-6)(17,6,-18,-7)(2,9,-3,-10)(18,11,-13,-12)(4,13,-5,-14)(14,3,-15,-4)(8,15,-9,-16)(1,16,-2,-17)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-17,-7)(-2,-10,-6,17)(-3,14,-5,10)(-4,-14)(-8,-16,1)(-9,2,16)(-11,18,6)(-12,7,-18)(-13,4,-15,8,12)(3,9,15)(5,13,11)
Multiloop annotated with half-edges
11^2_50 annotated with half-edges