$12^{2}_{51}$ - Minimal pinning sets
Pinning sets for 12^2_51 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_51 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 482
of which optimal: 2
of which minimal: 11
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05126
on average over minimal pinning sets: 2.52727
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 5, 7}
4
[2, 2, 2, 4]
2.50
B (optimal)
• {2, 5, 7, 9}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {1, 2, 5, 7, 10}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 2, 3, 5, 7}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {2, 5, 7, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
d (minimal)
• {2, 3, 5, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
e (minimal)
• {1, 2, 5, 6, 7}
5
[2, 2, 2, 3, 4]
2.60
f (minimal)
• {2, 5, 6, 7, 11}
5
[2, 2, 2, 4, 4]
2.80
g (minimal)
• {2, 5, 7, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
h (minimal)
• {2, 3, 5, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
i (minimal)
• {2, 5, 6, 7, 8}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.5
5
0
9
15
2.67
6
0
0
76
2.88
7
0
0
124
3.01
8
0
0
126
3.11
9
0
0
84
3.19
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
9
471
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,6],[0,6,7,8],[0,5,1,1],[1,4,9,2],[2,9,3,2],[3,9,8,8],[3,7,7,9],[5,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,20,3,11],[13,9,14,10],[4,8,5,7],[1,12,2,11],[12,19,13,20],[8,14,9,15],[5,18,6,17],[6,16,7,17],[18,15,19,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(14,7,-15,-8)(1,8,-2,-9)(6,15,-7,-16)(19,16,-20,-17)(4,17,-5,-18)(18,3,-19,-4)(13,20,-14,-11)(10,11,-1,-12)(12,9,-13,-10)
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,-9,12)(-2,5,17,-20,13,9)(-3,18,-5)(-4,-18)(-6,-16,19,3)(-7,14,20,16)(-8,1,11,-14)(-10,-12)(-11,10,-13)(-15,6,2,8)(-17,4,-19)(7,15)
Multiloop annotated with half-edges
12^2_51 annotated with half-edges