$12^{2}_{138}$ - Minimal pinning sets
Pinning sets for 12^2_138 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_138 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
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: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 8}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 7, 9}
5
[2, 2, 2, 2, 3]
2.20
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
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,8,8],[0,8,9,9],[0,9,9,1],[1,8,7,6],[1,5,7,7],[2,6,6,5],[2,5,3,2],[3,4,4,3]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,17,8,16],[5,13,6,14],[19,3,20,4],[1,18,2,17],[8,12,9,11],[15,10,16,11],[14,10,15,9],[12,4,13,5],[2,18,3,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(15,2,-16,-3)(12,17,-13,-18)(18,13,-19,-14)(14,11,-15,-12)(3,16,-4,-17)(19,10,-20,-11)(20,5,-7,-6)(6,7,-1,-8)(4,9,-5,-10)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,15,11,-20,-6,-8)(-3,-17,12,-15)(-4,-10,19,13,17)(-5,20,10)(-7,6)(-9,4,16,2)(-11,14,-19)(-12,-18,-14)(-13,18)(-16,3)(1,7,5,9)
Multiloop annotated with half-edges
12^2_138 annotated with half-edges