$11^{2}_{17}$ - Minimal pinning sets
Pinning sets for 11^2_17 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_17 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 222
of which optimal: 8
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.06096
on average over minimal pinning sets: 2.71
on average over optimal pinning sets: 2.7
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 6, 7, 8}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {2, 4, 6, 7, 8}
5
[2, 2, 3, 3, 4]
2.80
C (optimal)
• {2, 4, 5, 7, 8}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {2, 6, 7, 8, 10}
5
[2, 2, 3, 3, 4]
2.80
E (optimal)
• {2, 3, 5, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
F (optimal)
• {2, 5, 6, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
G (optimal)
• {2, 4, 5, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
H (optimal)
• {2, 6, 7, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 3, 5, 7, 8}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {2, 3, 5, 7, 8, 10}
6
[2, 2, 3, 3, 3, 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
8
0
0
2.7
6
0
2
36
2.89
7
0
0
69
3.04
8
0
0
64
3.13
9
0
0
33
3.2
10
0
0
9
3.24
11
0
0
1
3.27
Total
8
2
212
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,6,7],[0,5,1,1],[1,4,8,2],[2,8,7,3],[3,6,8,8],[5,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,18,3,11],[13,9,14,10],[4,14,5,15],[1,12,2,11],[12,17,13,18],[8,5,9,6],[15,8,16,7],[16,6,17,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,3,-7,-4)(17,4,-18,-5)(5,16,-6,-17)(14,7,-15,-8)(1,8,-2,-9)(2,15,-3,-16)(13,18,-14,-11)(10,11,-1,-12)(12,9,-13,-10)
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,-9,12)(-2,-16,5,-18,13,9)(-3,6,16)(-4,17,-6)(-5,-17)(-7,14,18,4)(-8,1,11,-14)(-10,-12)(-11,10,-13)(-15,2,8)(3,15,7)
Multiloop annotated with half-edges
11^2_17 annotated with half-edges