$12^{2}_{205}$ - Minimal pinning sets
Pinning sets for 12^2_205 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_205 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 240
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: 3.03455
on average over minimal pinning sets: 2.45
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, 5, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 5, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 4, 9, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 4, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
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
13
2.66
7
0
0
45
2.86
8
0
0
71
3.02
9
0
0
64
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
2
236
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 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],[0,1,5,6],[0,7,8,9],[0,9,9,5],[1,4,2,1],[2,8,7,7],[3,6,6,8],[3,7,6,9],[3,8,4,4]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,8,12,7],[9,6,10,7],[19,3,20,4],[1,14,2,13],[8,13,9,12],[5,17,6,18],[4,17,5,16],[18,15,19,16],[2,14,3,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(7,4,-8,-5)(5,18,-6,-19)(19,6,-20,-7)(20,9,-11,-10)(10,11,-1,-12)(15,12,-16,-13)(13,2,-14,-3)(3,14,-4,-15)(8,17,-9,-18)
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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,12)(-2,13,-16)(-3,-15,-13)(-4,7,-20,-10,-12,15)(-5,-19,-7)(-6,19)(-8,-18,5)(-9,20,6,18)(-11,10)(-14,3)(-17,8,4,14,2)(1,11,9,17)
Multiloop annotated with half-edges
12^2_205 annotated with half-edges