$10^{2}_{20}$ - Minimal pinning sets
Pinning sets for 10^2_20 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_20 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 64
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91518
on average over minimal pinning sets: 2.53333
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 7}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 5, 7, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
b (minimal)
• {1, 2, 3, 6, 7, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.4
6
0
2
10
2.67
7
0
0
23
2.89
8
0
0
19
3.05
9
0
0
7
3.14
10
0
0
1
3.2
Total
2
2
60
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,4,5],[0,5,6,6],[1,7,7,2],[2,7,6,3],[3,5,7,3],[4,6,5,4]]
PD code (use to draw this multiloop with SnapPy): [[6,16,1,7],[7,5,8,6],[8,15,9,16],[1,12,2,11],[14,4,15,5],[9,13,10,12],[2,10,3,11],[3,13,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(11,2,-12,-3)(5,14,-6,-15)(15,4,-16,-5)(16,13,-7,-14)(6,7,-1,-8)(12,9,-13,-10)(3,10,-4,-11)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,11,-4,15,-6,-8)(-3,-11)(-5,-15)(-7,6,14)(-9,12,2)(-10,3,-12)(-13,16,4,10)(-14,5,-16)(1,7,13,9)
Multiloop annotated with half-edges
10^2_20 annotated with half-edges