$12^{2}_{270}$ - Minimal pinning sets
Pinning sets for 12^2_270 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_270 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 216
of which optimal: 1
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98842
on average over minimal pinning sets: 2.425
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 5, 6, 9}
5
[2, 2, 2, 2, 4]
2.40
a (minimal)
• {2, 3, 6, 7, 9, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 3, 6, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
c (minimal)
• {2, 3, 6, 7, 9, 11}
6
[2, 2, 2, 2, 3, 5]
2.67
d (minimal)
• {2, 3, 4, 6, 9, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
e (minimal)
• {2, 3, 4, 6, 8, 9}
6
[2, 2, 2, 2, 3, 5]
2.67
f (minimal)
• {2, 3, 4, 6, 7, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
g (minimal)
• {1, 2, 3, 4, 6, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.4
6
0
7
7
2.55
7
0
0
44
2.81
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
7
208
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,5],[0,6,3,0],[0,2,7,7],[1,7,8,5],[1,4,9,1],[2,9,9,8],[3,8,4,3],[4,7,6,9],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,6,16,5],[19,13,20,14],[1,13,2,12],[6,3,7,4],[16,4,17,5],[18,8,19,9],[2,11,3,12],[7,11,8,10],[17,10,18,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(15,2,-16,-3)(3,20,-4,-15)(17,6,-18,-7)(7,16,-8,-17)(8,5,-9,-6)(14,9,-1,-10)(10,13,-11,-14)(18,11,-19,-12)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,20,-13,10)(-2,15,-4)(-3,-15)(-5,8,16,2)(-6,17,-8)(-7,-17)(-9,14,-11,18,6)(-10,-14)(-12,-20,3,-16,7,-18)(-19,12)(1,9,5)(11,13,19)
Multiloop annotated with half-edges
12^2_270 annotated with half-edges