$12^{2}_{292}$ - Minimal pinning sets
Pinning sets for 12^2_292 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_292 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 380
of which optimal: 3
of which minimal: 11
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12095
on average over minimal pinning sets: 2.77273
on average over optimal pinning sets: 2.66667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 5, 6, 10, 11}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 4, 5, 9, 12}
5
[2, 2, 3, 3, 4]
2.80
C (optimal)
• {1, 3, 4, 5, 9}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 5, 6, 9, 11, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
b (minimal)
• {1, 4, 5, 9, 10, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 4, 5, 7, 10, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
d (minimal)
• {1, 3, 5, 6, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
e (minimal)
• {1, 2, 5, 6, 9, 12}
6
[2, 2, 3, 3, 4, 4]
3.00
f (minimal)
• {1, 2, 5, 6, 9, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
g (minimal)
• {1, 2, 4, 5, 9, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
h (minimal)
• {1, 2, 3, 5, 6, 9}
6
[2, 2, 3, 3, 3, 4]
2.83
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
2.67
6
0
8
20
2.85
7
0
0
83
3.01
8
0
0
119
3.13
9
0
0
94
3.21
10
0
0
42
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
3
8
369
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,6,7,7],[0,8,8,1],[1,8,9,2],[2,9,7,3],[3,6,9,3],[4,9,5,4],[5,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,10,16,9],[13,8,14,9],[5,19,6,20],[1,11,2,10],[16,12,17,13],[4,7,5,8],[18,6,19,7],[11,3,12,2],[17,3,18,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,14,-16,-1)(16,3,-17,-4)(1,4,-2,-5)(10,5,-11,-6)(6,19,-7,-20)(7,12,-8,-13)(20,9,-15,-10)(13,8,-14,-9)(2,17,-3,-18)(11,18,-12,-19)
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,-5,10,-15)(-2,-18,11,5)(-3,16,14,8,12,18)(-4,1,-16)(-6,-20,-10)(-7,-13,-9,20)(-8,13)(-11,-19,6)(-12,7,19)(-14,15,9)(-17,2,4)(3,17)
Multiloop annotated with half-edges
12^2_292 annotated with half-edges