$12^{2}_{251}$ - Minimal pinning sets
Pinning sets for 12^2_251 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_251 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 104
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97397
on average over minimal pinning sets: 2.40476
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, 3, 5, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
a (minimal)
• {1, 2, 3, 5, 7, 9, 12}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
b (minimal)
• {1, 2, 3, 6, 7, 8, 9}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
c (minimal)
• {1, 2, 3, 6, 7, 9, 12}
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
1
0
0
2.33
7
0
3
6
2.56
8
0
0
26
2.81
9
0
0
35
3.01
10
0
0
24
3.17
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
3
100
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: [[1,2,2,3],[0,4,5,6],[0,7,7,0],[0,8,9,9],[1,9,8,5],[1,4,8,6],[1,5,7,7],[2,6,6,2],[3,5,4,9],[3,8,4,3]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,16,10,17],[19,7,20,8],[1,14,2,13],[3,15,4,16],[10,4,11,5],[17,5,18,6],[6,18,7,19],[14,11,15,12],[2,12,3,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(11,2,-12,-3)(13,4,-14,-5)(20,17,-9,-18)(7,18,-8,-19)(19,6,-20,-7)(8,9,-1,-10)(15,10,-16,-11)(3,12,-4,-13)(5,14,-6,-15)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,10)(-2,11,-16)(-3,-13,-5,-15,-11)(-4,13)(-6,19,-8,-10,15)(-7,-19)(-9,8,18)(-12,3)(-14,5)(-17,20,6,14,4,12,2)(-18,7,-20)(1,9,17)
Multiloop annotated with half-edges
12^2_251 annotated with half-edges