$11^{2}_{40}$ - Minimal pinning sets
Pinning sets for 11^2_40 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_40 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 188
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97587
on average over minimal pinning sets: 2.49
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 4, 6, 10}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 3, 4, 6, 11}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
d (minimal)
• {1, 3, 4, 6, 8}
5
[2, 2, 2, 3, 5]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
4
7
2.56
6
0
0
35
2.8
7
0
0
55
2.96
8
0
0
50
3.08
9
0
0
27
3.16
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
4
183
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,6],[0,6,7,4],[1,3,7,8],[2,8,6,6],[2,5,5,3],[3,8,8,4],[4,7,7,5]]
PD code (use to draw this multiloop with SnapPy): [[12,18,1,13],[13,11,14,12],[14,17,15,18],[1,6,2,7],[7,10,8,11],[4,16,5,17],[15,5,16,6],[2,9,3,10],[8,3,9,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(18,3,-13,-4)(11,4,-12,-5)(12,7,-1,-8)(5,8,-6,-9)(16,9,-17,-10)(2,13,-3,-14)(17,14,-18,-15)(10,15,-11,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,8)(-2,-14,17,9,-6)(-3,18,14)(-4,11,15,-18)(-5,-9,16,-11)(-7,12,4,-13,2)(-8,5,-12)(-10,-16)(-15,10,-17)(1,7)(3,13)
Multiloop annotated with half-edges
11^2_40 annotated with half-edges