$12^{2}_{257}$ - Minimal pinning sets
Pinning sets for 12^2_257 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_257 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 120
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9731
on average over minimal pinning sets: 2.38095
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {2, 3, 4, 5, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {1, 2, 4, 5, 6, 9, 10}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
b (minimal)
• {2, 3, 4, 5, 6, 9, 10}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.33
7
0
2
11
2.6
8
0
0
32
2.84
9
0
0
39
3.03
10
0
0
25
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
2
116
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,8,8,3],[0,2,4,4],[0,3,3,9],[1,9,9,6],[1,5,9,7],[1,6,8,8],[2,7,7,2],[4,6,5,5]]
PD code (use to draw this multiloop with SnapPy): [[12,3,1,4],[4,13,5,20],[11,17,12,18],[2,16,3,17],[1,16,2,15],[13,7,14,8],[5,8,6,9],[9,19,10,20],[18,10,19,11],[6,14,7,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(18,5,-19,-6)(3,6,-4,-7)(15,8,-16,-9)(20,11,-13,-12)(12,13,-1,-14)(9,14,-10,-15)(7,16,-8,-17)(17,2,-18,-3)(4,19,-5,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,14)(-2,17,-8,15,-10)(-3,-7,-17)(-4,-20,-12,-14,9,-16,7)(-5,18,2,-11,20)(-6,3,-18)(-9,-15)(-13,12)(-19,4,6)(1,13,11)(5,19)(8,16)
Multiloop annotated with half-edges
12^2_257 annotated with half-edges