$12^{2}_{324}$ - Minimal pinning sets
Pinning sets for 12^2_324 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_324 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 376
of which optimal: 7
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.06695
on average over minimal pinning sets: 2.5875
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {4, 5, 7, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
B (optimal)
• {4, 5, 6, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 4, 5, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {3, 4, 8, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {3, 4, 6, 9, 11}
5
[2, 2, 2, 4, 4]
2.80
F (optimal)
• {3, 4, 5, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
G (optimal)
• {2, 4, 6, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 2, 4, 8, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
7
0
0
2.6
6
0
1
38
2.82
7
0
0
90
2.98
8
0
0
112
3.09
9
0
0
82
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
7
1
368
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,7,8],[0,8,8,9],[0,9,7,1],[1,6,6,1],[2,5,5,7],[2,6,4,9],[2,9,3,3],[3,8,7,4]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,14,6,15],[17,3,18,4],[19,10,20,11],[1,13,2,14],[6,16,7,15],[7,16,8,17],[8,2,9,3],[18,12,19,11],[12,9,13,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(14,1,-15,-2)(9,6,-10,-7)(16,7,-17,-8)(8,15,-9,-16)(3,10,-4,-11)(18,11,-19,-12)(12,19,-13,-20)(20,13,-5,-14)(2,17,-3,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-5)(-2,-18,-12,-20,-14)(-3,-11,18)(-4,5,13,19,11)(-6,9,15,1)(-7,16,-9)(-8,-16)(-10,3,17,7)(-13,20)(-15,8,-17,2)(-19,12)(4,10,6)
Multiloop annotated with half-edges
12^2_324 annotated with half-edges