$12^{2}_{267}$ - Minimal pinning sets
Pinning sets for 12^2_267 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_267 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 396
of which optimal: 6
of which minimal: 12
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.11219
on average over minimal pinning sets: 2.73889
on average over optimal pinning sets: 2.7
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 2, 4, 6, 7}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 2, 6, 7, 12}
5
[2, 2, 3, 3, 5]
3.00
D (optimal)
• {1, 2, 6, 7, 9}
5
[2, 2, 3, 3, 3]
2.60
E (optimal)
• {1, 2, 6, 7, 10}
5
[2, 2, 3, 3, 4]
2.80
F (optimal)
• {1, 2, 6, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 4, 5, 8, 9}
6
[2, 2, 3, 3, 3, 4]
2.83
b (minimal)
• {1, 2, 4, 5, 7, 8}
6
[2, 2, 3, 3, 3, 4]
2.83
c (minimal)
• {1, 2, 4, 5, 9, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
d (minimal)
• {1, 2, 4, 5, 7, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
e (minimal)
• {1, 2, 4, 5, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
f (minimal)
• {1, 2, 4, 5, 7, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
6
0
0
2.7
6
0
6
31
2.87
7
0
0
91
3.02
8
0
0
119
3.13
9
0
0
91
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
6
6
384
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],[0,5,6,2],[0,1,6,3],[0,2,7,7],[0,8,8,5],[1,4,9,6],[1,5,9,2],[3,9,8,3],[4,7,9,4],[5,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[14,5,1,6],[6,15,7,20],[13,19,14,20],[4,18,5,19],[1,10,2,11],[15,11,16,12],[7,12,8,13],[17,3,18,4],[9,2,10,3],[16,9,17,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,14,-8,-1)(20,1,-15,-2)(12,3,-13,-4)(6,19,-7,-20)(13,8,-14,-9)(18,9,-19,-10)(5,10,-6,-11)(2,15,-3,-16)(11,16,-12,-17)(17,4,-18,-5)
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,-7)(-2,-16,11,-6,-20)(-3,12,16)(-4,17,-12)(-5,-11,-17)(-8,13,3,15,1)(-9,18,4,-13)(-10,5,-18)(-14,7,19,9)(-15,2)(-19,6,10)(8,14)
Multiloop annotated with half-edges
12^2_267 annotated with half-edges