$12^{3}_{56}$ - Minimal pinning sets
Pinning sets for 12^3_56 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_56 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 324
of which optimal: 2
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.106
on average over minimal pinning sets: 2.72593
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, 3, 5, 7, 10}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 2, 5, 9, 12}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 3, 5, 6, 9, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
b (minimal)
• {1, 3, 4, 5, 9, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
c (minimal)
• {1, 2, 3, 5, 9, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
d (minimal)
• {1, 2, 5, 7, 10, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
e (minimal)
• {1, 3, 5, 7, 9, 12}
6
[2, 2, 3, 3, 3, 3]
2.67
f (minimal)
• {1, 3, 5, 6, 9, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
g (minimal)
• {1, 3, 4, 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
2
0
0
2.6
6
0
7
14
2.79
7
0
0
66
2.97
8
0
0
100
3.1
9
0
0
84
3.2
10
0
0
40
3.26
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
7
315
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: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,3],[0,2,7,7],[0,7,8,8],[1,8,9,6],[1,5,9,2],[3,9,4,3],[4,9,5,4],[5,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[6,14,1,7],[7,15,8,20],[5,19,6,20],[13,18,14,19],[1,12,2,11],[15,3,16,4],[8,4,9,5],[17,12,18,13],[2,10,3,11],[16,10,17,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-15,-2)(13,2,-14,-3)(14,15,-7,-16)(6,7,-1,-8)(8,5,-9,-6)(16,9,-17,-10)(4,11,-5,-12)(10,17,-11,-18)(3,18,-4,-19)(19,12,-20,-13)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,12,-5,8)(-2,13,-20)(-3,-19,-13)(-4,-12,19)(-6,-8)(-7,6,-9,16)(-10,-18,3,-14,-16)(-11,4,18)(-15,14,2)(-17,10)(1,7,15)(5,11,17,9)
Multiloop annotated with half-edges
12^3_56 annotated with half-edges