$12^{2}_{47}$ - Minimal pinning sets
Pinning sets for 12^2_47 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_47 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97043
on average over minimal pinning sets: 2.26667
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, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,7,8],[0,9,9,6],[0,5,5,1],[1,4,4,6],[1,5,3,7],[2,6,8,2],[2,7,9,9],[3,8,8,3]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,10,8,11],[5,15,6,16],[19,12,20,13],[1,9,2,10],[8,2,9,3],[11,3,12,4],[16,4,17,5],[17,14,18,15],[13,18,14,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(13,4,-14,-5)(16,5,-17,-6)(3,14,-4,-15)(12,15,-13,-16)(10,17,-11,-18)(8,19,-9,-20)(1,20,-2,-7)(18,9,-19,-10)(2,11,-3,-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)(-2,-12,-16,-6,7)(-3,-15,12)(-4,13,15)(-5,16,-13)(-8,-20,1)(-9,18,-11,2,20)(-10,-18)(-14,3,11,17,5)(-17,10,-19,8,6)(4,14)(9,19)
Multiloop annotated with half-edges
12^2_47 annotated with half-edges