$12^{2}_{294}$ - Minimal pinning sets
Pinning sets for 12^2_294 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_294 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 372
of which optimal: 4
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10651
on average over minimal pinning sets: 2.71111
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)
• {1, 5, 6, 7, 12}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 3, 7, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 3, 6, 7, 12}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 3, 6, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 5, 7, 8, 11, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 4, 5, 7, 8, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
c (minimal)
• {1, 3, 4, 7, 8, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
d (minimal)
• {1, 2, 5, 7, 8, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
e (minimal)
• {1, 2, 3, 7, 8, 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
4
0
0
2.6
6
0
5
25
2.82
7
0
0
82
3.0
8
0
0
114
3.12
9
0
0
90
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
4
5
363
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],[0,1,6,6],[0,7,8,8],[0,9,5,1],[1,4,9,6],[2,5,7,2],[3,6,9,8],[3,7,9,3],[4,8,7,5]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,6,16,5],[13,4,14,5],[9,19,10,20],[1,7,2,6],[16,2,17,3],[3,12,4,13],[8,11,9,12],[18,10,19,11],[7,18,8,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,14,-16,-1)(11,2,-12,-3)(3,20,-4,-15)(4,13,-5,-14)(17,8,-18,-9)(6,9,-7,-10)(1,10,-2,-11)(7,18,-8,-19)(12,19,-13,-20)(16,5,-17,-6)
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,-11,-3,-15)(-2,11)(-4,-14,15)(-5,16,14)(-6,-10,1,-16)(-7,-19,12,2,10)(-8,17,5,13,19)(-9,6,-17)(-12,-20,3)(-13,4,20)(-18,7,9)(8,18)
Multiloop annotated with half-edges
12^2_294 annotated with half-edges