$11^{3}_{16}$ - Minimal pinning sets
Pinning sets for 11^3_16 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^3_16 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 92
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: 2.91244
on average over minimal pinning sets: 2.38333
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 7, 10, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 7, 8, 10}
6
[2, 2, 2, 2, 3, 4]
2.50
b (minimal)
• {1, 2, 3, 6, 7, 10}
6
[2, 2, 2, 2, 3, 4]
2.50
c (minimal)
• {1, 2, 3, 5, 7, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
3
6
2.5
7
0
0
24
2.77
8
0
0
30
2.97
9
0
0
20
3.1
10
0
0
7
3.2
11
0
0
1
3.27
Total
1
3
88
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,3,3],[0,2,2,6],[0,6,7,7],[1,7,8,1],[3,8,8,4],[4,8,5,4],[5,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[5,8,6,1],[4,14,5,9],[7,13,8,14],[6,13,7,12],[1,15,2,18],[9,3,10,4],[11,15,12,16],[2,17,3,18],[10,17,11,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(3,6,-4,-7)(17,4,-18,-5)(5,18,-6,-13)(2,13,-3,-14)(12,15,-9,-16)(8,9,-1,-10)(10,7,-11,-8)(16,11,-17,-12)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-3,-7,10)(-2,-14)(-4,17,11,7)(-5,-13,2,-15,12,-17)(-6,3,13)(-8,-10)(-9,8,-11,16)(-12,-16)(-18,5)(1,9,15)(4,6,18)
Multiloop annotated with half-edges
11^3_16 annotated with half-edges