$12^{2}_{168}$ - Minimal pinning sets
Pinning sets for 12^2_168 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_168 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 5, 8, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {2, 3, 6, 8, 9, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,6],[0,7,7,8],[0,8,9,9],[1,9,8,1],[2,7,7,2],[3,6,6,3],[3,5,9,4],[4,8,5,4]]
PD code (use to draw this multiloop with SnapPy): [[5,10,6,1],[4,20,5,11],[9,19,10,20],[6,17,7,16],[1,13,2,14],[11,3,12,4],[18,8,19,9],[17,8,18,7],[12,15,13,16],[2,15,3,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (3,18,-4,-19)(19,4,-20,-5)(1,6,-2,-7)(12,9,-13,-10)(7,14,-8,-15)(15,8,-16,-9)(13,16,-14,-17)(17,2,-18,-3)(5,20,-6,-11)(10,11,-1,-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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-7,-15,-9,12)(-2,17,-14,7)(-3,-19,-5,-11,10,-13,-17)(-4,19)(-6,1,11)(-8,15)(-10,-12)(-16,13,9)(-18,3)(-20,5)(2,6,20,4,18)(8,14,16)
Multiloop annotated with half-edges
12^2_168 annotated with half-edges