$12^{2}_{140}$ - Minimal pinning sets
Pinning sets for 12^2_140 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_140 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, 3, 5, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 6, 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: [[1,2,2,3],[0,3,4,4],[0,4,5,0],[0,6,7,1],[1,8,2,1],[2,8,8,6],[3,5,9,7],[3,6,9,9],[4,9,5,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,11,18,12],[19,15,20,16],[1,10,2,11],[18,13,19,12],[5,14,6,15],[6,9,7,10],[2,7,3,8],[13,4,14,5],[8,3,9,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(11,2,-12,-3)(16,5,-1,-6)(4,7,-5,-8)(15,8,-16,-9)(20,9,-17,-10)(10,17,-11,-18)(3,12,-4,-13)(18,13,-19,-14)(14,19,-15,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6)(-2,11,17,9,-16,-6)(-3,-13,18,-11)(-4,-8,15,19,13)(-5,16,8)(-7,4,12,2)(-9,20,-15)(-10,-18,-14,-20)(-12,3)(-17,10)(-19,14)(1,5,7)
Multiloop annotated with half-edges
12^2_140 annotated with half-edges