$12^{2}_{318}$ - Minimal pinning sets
Pinning sets for 12^2_318 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_318 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 336
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04109
on average over minimal pinning sets: 2.49444
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)
• {1, 3, 4, 10}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 4, 5, 9}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 3, 4, 8, 9, 12}
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
1
8
2.56
6
0
1
34
2.78
7
0
0
75
2.94
8
0
0
96
3.07
9
0
0
75
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
1
2
333
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,8,8,3],[0,2,6,4],[0,3,5,5],[1,4,4,6],[1,5,3,9],[1,9,9,8],[2,7,9,2],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,3,18,4],[15,10,16,11],[19,9,20,10],[1,9,2,8],[2,7,3,8],[18,7,19,6],[4,13,5,14],[11,14,12,15],[12,5,13,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-17,-2)(10,3,-11,-4)(4,17,-5,-18)(16,5,-1,-6)(6,15,-7,-16)(18,7,-19,-8)(13,8,-14,-9)(2,11,-3,-12)(9,12,-10,-13)(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,-15,6)(-2,-12,9,-14,-20)(-3,10,12)(-4,-18,-8,13,-10)(-5,16,-7,18)(-6,-16)(-9,-13)(-11,2,-17,4)(-19,14,8)(1,5,17)(3,11)(7,15,19)
Multiloop annotated with half-edges
12^2_318 annotated with half-edges