$12^{2}_{62}$ - Minimal pinning sets
Pinning sets for 12^2_62 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_62 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 400
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: 3.10335
on average over minimal pinning sets: 2.59167
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, 8, 9}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 3, 6, 9, 10}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 2, 5, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 2, 5, 6, 9, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.5
5
0
2
8
2.72
6
0
1
40
2.89
7
0
0
90
3.02
8
0
0
116
3.12
9
0
0
90
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
3
396
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,7,5,8],[0,8,8,4],[0,3,8,1],[1,2,9,6],[1,5,9,7],[2,6,9,9],[2,4,3,3],[5,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,5,14,6],[17,11,18,12],[3,19,4,20],[1,4,2,5],[14,11,15,10],[6,10,7,9],[16,8,17,9],[18,2,19,3],[15,8,16,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,12,-14,-1)(11,2,-12,-3)(16,3,-17,-4)(5,18,-6,-19)(19,6,-20,-7)(7,4,-8,-5)(8,17,-9,-18)(20,9,-13,-10)(1,14,-2,-15)(10,15,-11,-16)
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,-15,10,-13)(-2,11,15)(-3,16,-11)(-4,7,-20,-10,-16)(-5,-19,-7)(-6,19)(-8,-18,5)(-9,20,6,18)(-12,13,9,17,3)(-14,1)(-17,8,4)(2,14,12)
Multiloop annotated with half-edges
12^2_62 annotated with half-edges