$12^{2}_{253}$ - Minimal pinning sets
Pinning sets for 12^2_253 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_253 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 184
of which optimal: 5
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04022
on average over minimal pinning sets: 2.53968
on average over optimal pinning sets: 2.53333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 7, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
B (optimal)
• {1, 3, 5, 7, 11, 12}
6
[2, 2, 2, 3, 3, 3]
2.50
C (optimal)
• {1, 3, 5, 8, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
D (optimal)
• {1, 3, 4, 5, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
E (optimal)
• {1, 2, 4, 5, 9, 11}
6
[2, 2, 2, 3, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 5, 7, 11, 12}
7
[2, 2, 2, 3, 3, 3, 3]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
5
0
0
2.53
7
0
1
25
2.78
8
0
0
54
2.97
9
0
0
57
3.12
10
0
0
32
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
5
1
178
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,3],[0,2,6,6],[0,6,7,8],[1,9,2,1],[3,7,4,3],[4,6,9,8],[4,7,9,9],[5,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,10,14,9],[11,8,12,9],[19,7,20,8],[1,18,2,17],[10,15,11,14],[6,18,7,19],[2,6,3,5],[16,4,17,5],[15,4,16,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,13,-1,-14)(14,1,-15,-2)(2,11,-3,-12)(17,4,-18,-5)(8,5,-9,-6)(6,19,-7,-20)(15,10,-16,-11)(3,16,-4,-17)(9,18,-10,-19)(20,7,-13,-8)
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)(-2,-12,-14)(-3,-17,-5,8,-13,12)(-4,17)(-6,-20,-8)(-7,20)(-9,-19,6)(-10,15,1,13,7,19)(-11,2,-15)(-16,3,11)(-18,9,5)(4,16,10,18)
Multiloop annotated with half-edges
12^2_253 annotated with half-edges