$12^{2}_{310}$ - Minimal pinning sets
Pinning sets for 12^2_310 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_310 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 436
of which optimal: 1
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05799
on average over minimal pinning sets: 2.63542
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 6, 8, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {2, 6, 7, 9, 10}
5
[2, 2, 2, 4, 4]
2.80
b (minimal)
• {2, 6, 7, 9, 11}
5
[2, 2, 2, 4, 4]
2.80
c (minimal)
• {2, 4, 6, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
d (minimal)
• {2, 4, 6, 9, 10}
5
[2, 2, 2, 3, 4]
2.60
e (minimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
f (minimal)
• {2, 5, 6, 9, 10}
5
[2, 2, 2, 4, 4]
2.80
g (minimal)
• {1, 2, 5, 6, 9, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
6
8
2.61
6
0
1
56
2.84
7
0
0
111
3.0
8
0
0
123
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
1
7
428
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,6,6,3],[0,2,7,4],[1,3,7,8],[1,8,9,6],[2,5,7,2],[3,6,9,4],[4,9,9,5],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,11,14,12],[19,7,20,8],[1,7,2,6],[10,5,11,6],[14,17,15,18],[8,18,9,19],[2,9,3,10],[16,4,17,5],[15,4,16,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,13,-1,-14)(14,1,-15,-2)(7,2,-8,-3)(3,10,-4,-11)(17,4,-18,-5)(11,6,-12,-7)(19,8,-20,-9)(20,15,-13,-16)(5,16,-6,-17)(9,18,-10,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14)(-2,7,-12,-14)(-3,-11,-7)(-4,17,-6,11)(-5,-17)(-8,19,-10,3)(-9,-19)(-13,12,6,16)(-15,20,8,2)(-16,5,-18,9,-20)(1,13,15)(4,10,18)
Multiloop annotated with half-edges
12^2_310 annotated with half-edges