$12^{2}_{295}$ - Minimal pinning sets
Pinning sets for 12^2_295 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_295 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 416
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: 3.11404
on average over minimal pinning sets: 2.67917
on average over optimal pinning sets: 2.65714
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 7, 9}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 2, 3, 5, 11}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 2, 3, 4, 9}
5
[2, 2, 3, 3, 3]
2.60
E (optimal)
• {1, 2, 3, 4, 11}
5
[2, 2, 3, 3, 3]
2.60
F (optimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 3, 3, 4]
2.80
G (optimal)
• {1, 2, 4, 6, 11}
5
[2, 2, 3, 3, 4]
2.80
a (minimal)
• {1, 2, 3, 5, 9, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
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
2.66
6
0
1
39
2.87
7
0
0
96
3.03
8
0
0
125
3.14
9
0
0
95
3.22
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
7
1
408
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,6,7,7],[0,8,9,1],[1,9,9,2],[2,8,7,3],[3,6,8,3],[4,7,6,9],[4,8,5,5]]
PD code (use to draw this multiloop with SnapPy): [[5,12,6,1],[4,20,5,13],[11,19,12,20],[6,16,7,17],[1,14,2,13],[10,3,11,4],[15,18,16,19],[7,18,8,17],[14,8,15,9],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,12,-8,-1)(18,5,-19,-6)(11,6,-12,-7)(2,9,-3,-10)(3,20,-4,-13)(13,4,-14,-5)(19,14,-20,-15)(8,15,-9,-16)(1,16,-2,-17)(17,10,-18,-11)
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,-17,-11,-7)(-2,-10,17)(-3,-13,-5,18,10)(-4,13)(-6,11,-18)(-8,-16,1)(-9,2,16)(-12,7)(-14,19,5)(-15,8,12,6,-19)(-20,3,9,15)(4,20,14)
Multiloop annotated with half-edges
12^2_295 annotated with half-edges