$11^{2}_{18}$ - Minimal pinning sets
Pinning sets for 11^2_18 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_18 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 320
of which optimal: 3
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.06738
on average over minimal pinning sets: 2.75
on average over optimal pinning sets: 2.58333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 6, 8}
4
[2, 2, 3, 4]
2.75
B (optimal)
• {2, 5, 6, 9}
4
[2, 2, 3, 3]
2.50
C (optimal)
• {2, 3, 6, 9}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 5, 6, 8}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 2, 3, 6, 8}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {2, 4, 6, 9, 10}
5
[2, 2, 3, 4, 5]
3.20
d (minimal)
• {2, 4, 6, 7, 9}
5
[2, 2, 3, 4, 4]
3.00
e (minimal)
• {2, 5, 6, 8, 11}
5
[2, 2, 3, 3, 4]
2.80
f (minimal)
• {2, 3, 6, 8, 11}
5
[2, 2, 3, 3, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.58
5
0
6
20
2.82
6
0
0
72
2.97
7
0
0
98
3.08
8
0
0
76
3.15
9
0
0
35
3.21
10
0
0
9
3.24
11
0
0
1
3.27
Total
3
6
311
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,8],[0,5,1,1],[1,4,8,2],[2,8,7,7],[3,6,6,8],[3,7,6,5]]
PD code (use to draw this multiloop with SnapPy): [[3,8,4,1],[2,18,3,9],[11,7,12,8],[4,12,5,13],[1,10,2,9],[10,17,11,18],[14,6,15,7],[5,15,6,16],[13,16,14,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,4,-16,-5)(12,5,-13,-6)(1,6,-2,-7)(2,13,-3,-14)(3,16,-4,-17)(14,17,-15,-18)(11,18,-12,-9)(8,9,-1,-10)(10,7,-11,-8)
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,-7,10)(-2,-14,-18,11,7)(-3,-17,14)(-4,15,17)(-5,12,18,-15)(-6,1,9,-12)(-8,-10)(-9,8,-11)(-13,2,6)(-16,3,13,5)(4,16)
Multiloop annotated with half-edges
11^2_18 annotated with half-edges