$12^{2}_{288}$ - Minimal pinning sets
Pinning sets for 12^2_288 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_288 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 336
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: 3.10243
on average over minimal pinning sets: 2.625
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 12}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 5, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 2, 4, 5, 11, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 3, 4, 5, 9, 11}
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
4
1
0
0
2.5
5
0
0
8
2.75
6
0
3
28
2.89
7
0
0
70
3.01
8
0
0
96
3.11
9
0
0
80
3.19
10
0
0
39
3.26
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
3
332
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,6],[0,7,5,1],[1,4,7,2],[3,8,9,3],[4,9,8,5],[6,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,12,6,13],[13,3,14,4],[14,19,15,20],[1,11,2,12],[6,2,7,3],[18,15,19,16],[10,7,11,8],[16,10,17,9],[17,8,18,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(14,3,-15,-4)(9,16,-10,-17)(17,8,-18,-9)(2,19,-3,-20)(13,20,-14,-5)(10,7,-11,-8)(6,11,-7,-12)(1,12,-2,-13)(18,15,-19,-16)
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,-13,-5)(-2,-20,13)(-3,14,20)(-4,5,-14)(-6,-12,1)(-7,10,16,-19,2,12)(-8,17,-10)(-9,-17)(-11,6,4,-15,18,8)(-16,9,-18)(3,19,15)(7,11)
Multiloop annotated with half-edges
12^2_288 annotated with half-edges