$12^{2}_{289}$ - Minimal pinning sets
Pinning sets for 12^2_289 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_289 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 516
of which optimal: 2
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11721
on average over minimal pinning sets: 2.77778
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, 4, 6, 9}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 4, 5, 10}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 4, 6, 7, 10}
5
[2, 2, 3, 3, 5]
3.00
b (minimal)
• {1, 4, 6, 10, 11}
5
[2, 2, 3, 3, 5]
3.00
c (minimal)
• {1, 2, 3, 4, 6, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
d (minimal)
• {1, 4, 5, 8, 9, 12}
6
[2, 2, 3, 3, 4, 4]
3.00
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
2
16
2.78
6
0
2
64
2.94
7
0
0
128
3.06
8
0
0
146
3.16
9
0
0
102
3.22
10
0
0
43
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
4
510
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],[0,5,6,7],[0,8,9,3],[0,2,6,4],[0,3,5,5],[1,4,4,6],[1,5,3,7],[1,6,9,8],[2,7,9,9],[2,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,3,18,4],[15,12,16,13],[19,11,20,12],[1,11,2,10],[2,9,3,10],[18,9,19,8],[4,8,5,7],[13,7,14,6],[14,5,15,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-17,-2)(12,3,-13,-4)(4,11,-5,-12)(5,2,-6,-3)(6,17,-7,-18)(16,7,-1,-8)(8,15,-9,-16)(18,9,-19,-10)(13,10,-14,-11)(14,19,-15,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,-15,8)(-2,5,11,-14,-20)(-3,12,-5)(-4,-12)(-6,-18,-10,13,3)(-7,16,-9,18)(-8,-16)(-11,4,-13)(-17,6,2)(-19,14,10)(1,7,17)(9,15,19)
Multiloop annotated with half-edges
12^2_289 annotated with half-edges