$12^{2}_{314}$ - Minimal pinning sets
Pinning sets for 12^2_314 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_314 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 432
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11161
on average over minimal pinning sets: 2.66667
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)
• {1, 2, 7, 8}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 3, 6, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 2, 6, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 2, 6, 7, 9}
5
[2, 2, 3, 3, 4]
2.80
d (minimal)
• {1, 3, 4, 7, 8, 11}
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
4
1
0
0
2.5
5
0
3
8
2.73
6
0
1
45
2.9
7
0
0
100
3.04
8
0
0
126
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
1
4
427
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,6],[0,7,8,9],[0,9,4,0],[1,3,9,5],[1,4,9,8],[1,8,7,7],[2,6,6,8],[2,7,6,5],[2,5,4,3]]
PD code (use to draw this multiloop with SnapPy): [[5,14,6,1],[4,20,5,15],[13,17,14,18],[6,2,7,1],[15,7,16,8],[8,3,9,4],[19,11,20,12],[18,11,19,10],[12,9,13,10],[16,2,17,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,14,-16,-1)(6,1,-7,-2)(20,5,-15,-6)(10,7,-11,-8)(8,17,-9,-18)(18,9,-19,-10)(3,12,-4,-13)(13,4,-14,-5)(11,16,-12,-17)(2,19,-3,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-15)(-2,-20,-6)(-3,-13,-5,20)(-4,13)(-7,10,-19,2)(-8,-18,-10)(-9,18)(-11,-17,8)(-12,3,19,9,17)(-14,15,5)(-16,11,7,1)(4,12,16,14)
Multiloop annotated with half-edges
12^2_314 annotated with half-edges