$12^{2}_{216}$ - Minimal pinning sets
Pinning sets for 12^2_216 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_216 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 188
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97456
on average over minimal pinning sets: 2.40667
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 7}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 5, 7, 11}
6
[2, 2, 2, 2, 3, 3]
2.33
b (minimal)
• {1, 2, 4, 5, 7, 9}
6
[2, 2, 2, 2, 3, 4]
2.50
c (minimal)
• {1, 2, 4, 5, 7, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
d (minimal)
• {1, 2, 4, 5, 7, 12}
6
[2, 2, 2, 2, 3, 5]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
4
7
2.5
7
0
0
35
2.77
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
4
183
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,6,6],[0,7,8,8],[1,9,9,5],[1,4,6,2],[2,5,7,2],[3,6,9,8],[3,7,9,3],[4,8,7,4]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,5,8,6],[19,10,20,11],[1,15,2,16],[4,12,5,13],[8,12,9,11],[9,18,10,19],[14,17,15,18],[2,17,3,16],[13,3,14,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(15,2,-16,-3)(13,18,-14,-19)(19,16,-20,-17)(1,20,-2,-7)(4,9,-5,-10)(10,5,-11,-6)(8,11,-9,-12)(17,12,-18,-13)(3,14,-4,-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,-7)(-2,15,-4,-10,-6,7)(-3,-15)(-5,10)(-8,-12,17,-20,1)(-9,4,14,18,12)(-11,8,6)(-13,-19,-17)(-14,3,-16,19)(-18,13)(2,20,16)(5,9,11)
Multiloop annotated with half-edges
12^2_216 annotated with half-edges