$12^{2}_{313}$ - Minimal pinning sets
Pinning sets for 12^2_313 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_313 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 432
of which optimal: 8
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11442
on average over minimal pinning sets: 2.675
on average over optimal pinning sets: 2.675
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 2, 5, 8, 12}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {1, 2, 5, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
E (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
F (optimal)
• {1, 2, 5, 7, 10}
5
[2, 2, 3, 3, 4]
2.80
G (optimal)
• {1, 2, 5, 7, 8}
5
[2, 2, 3, 3, 3]
2.60
H (optimal)
• {1, 2, 3, 5, 8}
5
[2, 2, 3, 3, 3]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
8
0
0
2.68
6
0
0
44
2.89
7
0
0
102
3.03
8
0
0
129
3.14
9
0
0
96
3.22
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
8
0
424
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,6],[0,6,7,8],[0,8,4,0],[1,3,8,5],[1,4,8,9],[1,9,7,2],[2,6,9,9],[2,5,4,3],[5,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[5,12,6,1],[4,20,5,13],[11,15,12,16],[6,2,7,1],[13,7,14,8],[8,3,9,4],[19,16,20,17],[10,18,11,19],[14,2,15,3],[9,18,10,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,12,-14,-1)(6,1,-7,-2)(20,5,-13,-6)(3,10,-4,-11)(11,4,-12,-5)(9,14,-10,-15)(18,15,-19,-16)(16,7,-17,-8)(8,17,-9,-18)(2,19,-3,-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,-13)(-2,-20,-6)(-3,-11,-5,20)(-4,11)(-7,16,-19,2)(-8,-18,-16)(-9,-15,18)(-10,3,19,15)(-12,13,5)(-14,9,17,7,1)(-17,8)(4,10,14,12)
Multiloop annotated with half-edges
12^2_313 annotated with half-edges