$10^{2}_{9}$ - Minimal pinning sets
Pinning sets for 10^2_9 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_9 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 86
of which optimal: 3
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.99319
on average over minimal pinning sets: 2.66667
on average over optimal pinning sets: 2.66667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 3, 4, 7, 10}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 3, 4, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
a (minimal)
• {1, 2, 3, 5, 7, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 3, 4, 5, 7, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 2, 3, 4, 9, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.67
6
0
3
15
2.81
7
0
0
32
2.99
8
0
0
24
3.1
9
0
0
8
3.17
10
0
0
1
3.2
Total
3
3
80
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,2,2],[0,1,1,3],[0,2,5,6],[0,6,7,1],[3,7,7,6],[3,5,7,4],[4,6,5,5]]
PD code (use to draw this multiloop with SnapPy): [[8,5,1,6],[6,9,7,16],[7,15,8,16],[4,14,5,15],[1,10,2,9],[3,11,4,12],[13,10,14,11],[2,13,3,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,3,-7,-4)(9,8,-10,-1)(2,11,-3,-12)(5,12,-6,-13)(13,4,-14,-5)(14,7,-15,-8)(10,15,-11,-16)(1,16,-2,-9)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-9)(-2,-12,5,-14,-8,9)(-3,6,12)(-4,13,-6)(-5,-13)(-7,14,4)(-10,-16,1)(-11,2,16)(-15,10,8)(3,11,15,7)
Multiloop annotated with half-edges
10^2_9 annotated with half-edges