$12^{2}_{180}$ - Minimal pinning sets
Pinning sets for 12^2_180 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_180 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 304
of which optimal: 5
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04294
on average over minimal pinning sets: 2.48
on average over optimal pinning sets: 2.48
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {3, 5, 7, 9, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {3, 5, 7, 10, 11}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {3, 5, 6, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
E (optimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
5
0
0
2.48
6
0
0
28
2.74
7
0
0
67
2.93
8
0
0
89
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
5
0
299
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: [[1,1,2,3],[0,4,5,0],[0,5,5,3],[0,2,6,4],[1,3,7,8],[1,8,2,2],[3,9,9,7],[4,6,9,8],[4,7,9,5],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,15,18,16],[19,7,20,8],[1,7,2,6],[14,5,15,6],[18,9,19,8],[2,12,3,11],[13,10,14,11],[4,9,5,10],[12,4,13,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(2,13,-3,-14)(18,3,-19,-4)(4,17,-5,-18)(16,5,-1,-6)(9,6,-10,-7)(7,14,-8,-15)(20,11,-17,-12)(15,8,-16,-9)(12,19,-13,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,6)(-2,-14,7,-10)(-3,18,-5,16,8,14)(-4,-18)(-6,9,-16)(-7,-15,-9)(-8,15)(-11,20,-13,2)(-12,-20)(-17,4,-19,12)(1,5,17,11)(3,13,19)
Multiloop annotated with half-edges
12^2_180 annotated with half-edges