$12^{2}_{166}$ - Minimal pinning sets
Pinning sets for 12^2_166 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_166 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)
• {1, 3, 4, 5, 6, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 4, 5, 6, 9}
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, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,6,6,7],[0,7,8,8],[1,8,7,5],[1,4,9,9],[2,9,9,2],[2,4,8,3],[3,7,4,3],[5,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,13,16,14],[7,19,8,20],[1,11,2,10],[3,12,4,13],[16,4,17,5],[18,6,19,7],[8,12,9,11],[2,9,3,10],[17,6,18,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,15,-1,-16)(16,1,-17,-2)(11,4,-12,-5)(5,10,-6,-11)(6,3,-7,-4)(12,7,-13,-8)(19,8,-20,-9)(20,13,-15,-14)(2,17,-3,-18)(9,18,-10,-19)
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)(-2,-18,9,-20,-14,-16)(-3,6,10,18)(-4,11,-6)(-5,-11)(-7,12,4)(-8,19,-10,5,-12)(-9,-19)(-13,20,8)(-15,14)(-17,2)(1,15,13,7,3,17)
Multiloop annotated with half-edges
12^2_166 annotated with half-edges