$12^{2}_{210}$ - Minimal pinning sets
Pinning sets for 12^2_210 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_210 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 312
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04121
on average over minimal pinning sets: 2.5
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
B (optimal)
• {1, 3, 6, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 5, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
E (optimal)
• {1, 2, 3, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 3, 6, 9, 11}
5
[2, 2, 2, 3, 5]
2.80
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
2.5
6
0
0
31
2.76
7
0
0
70
2.93
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
6
0
306
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,6,3],[0,2,7,4],[1,3,7,8],[1,6,6,2],[2,5,5,9],[3,9,8,4],[4,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,7,10,8],[19,16,20,17],[1,16,2,15],[6,14,7,15],[10,18,11,17],[11,18,12,19],[2,5,3,6],[3,13,4,14],[12,4,13,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-9,-2)(7,2,-8,-3)(17,4,-18,-5)(8,9,-1,-10)(3,10,-4,-11)(11,6,-12,-7)(15,12,-16,-13)(13,18,-14,-19)(19,14,-20,-15)(5,16,-6,-17)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,14,18,4,10)(-2,7,-12,15,-20)(-3,-11,-7)(-4,17,-6,11)(-5,-17)(-8,-10,3)(-9,8,2)(-13,-19,-15)(-14,19)(-16,5,-18,13)(1,9)(6,16,12)
Multiloop annotated with half-edges
12^2_210 annotated with half-edges