$11^{2}_{16}$ - Minimal pinning sets
Pinning sets for 11^2_16 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_16 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 184
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97765
on average over minimal pinning sets: 2.4625
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)
• {2, 5, 7, 8}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {2, 5, 6, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {2, 4, 5, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
• {2, 3, 5, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
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
3
7
2.56
6
0
0
33
2.79
7
0
0
54
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
3
180
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: [[1,2,3,4],[0,4,4,5],[0,5,6,6],[0,6,7,8],[0,5,1,1],[1,4,8,2],[2,7,3,2],[3,6,8,8],[3,7,7,5]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,18,3,11],[13,9,14,10],[4,8,5,7],[1,12,2,11],[12,17,13,18],[8,14,9,15],[5,15,6,16],[16,6,17,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (17,4,-18,-5)(5,2,-6,-3)(14,7,-15,-8)(1,8,-2,-9)(6,15,-7,-16)(3,16,-4,-17)(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)(-2,5,-18,13,9)(-3,-17,-5)(-4,17)(-6,-16,3)(-7,14,18,4,16)(-8,1,11,-14)(-10,-12)(-11,10,-13)(-15,6,2,8)(7,15)
Multiloop annotated with half-edges
11^2_16 annotated with half-edges