$12^{2}_{60}$ - Minimal pinning sets
Pinning sets for 12^2_60 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_60 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 280
of which optimal: 3
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04505
on average over minimal pinning sets: 2.51111
on average over optimal pinning sets: 2.46667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 2, 3, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 5, 6, 9, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 3, 4, 6, 9, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
c (minimal)
• {1, 2, 3, 6, 9, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.47
6
0
3
18
2.71
7
0
0
58
2.9
8
0
0
84
3.05
9
0
0
70
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
3
3
274
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,7,0],[0,7,7,1],[1,6,8,5],[1,4,8,9],[2,9,4,7],[2,6,3,3],[4,9,9,5],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,12,6,13],[19,3,20,4],[1,11,2,12],[6,18,7,17],[13,17,14,16],[9,18,10,19],[10,2,11,3],[7,15,8,14],[8,15,9,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(14,1,-15,-2)(3,6,-4,-7)(16,7,-17,-8)(9,18,-10,-19)(19,10,-20,-11)(11,8,-12,-9)(12,17,-13,-18)(20,13,-5,-14)(2,15,-3,-16)
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,-5)(-2,-16,-8,11,-20,-14)(-3,-7,16)(-4,5,13,17,7)(-6,3,15,1)(-9,-19,-11)(-10,19)(-12,-18,9)(-13,20,10,18)(-15,2)(-17,12,8)(4,6)
Multiloop annotated with half-edges
12^2_60 annotated with half-edges