$12^{2}_{328}$ - Minimal pinning sets
Pinning sets for 12^2_328 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_328 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 388
of which optimal: 9
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.06099
on average over minimal pinning sets: 2.59
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 5, 6, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
B (optimal)
• {1, 4, 6, 9, 10}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 4, 6, 9, 11}
5
[2, 2, 2, 4, 4]
2.80
D (optimal)
• {1, 2, 3, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
E (optimal)
• {1, 2, 6, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
F (optimal)
• {1, 2, 6, 9, 12}
5
[2, 2, 2, 3, 4]
2.60
G (optimal)
• {1, 2, 6, 8, 9}
5
[2, 2, 2, 3, 5]
2.80
H (optimal)
• {1, 2, 5, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
I (optimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 3, 5, 6, 9, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
9
0
0
2.6
6
0
1
43
2.82
7
0
0
94
2.98
8
0
0
113
3.1
9
0
0
82
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
9
1
378
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,7,3],[0,2,8,9],[0,9,9,1],[1,8,6,1],[2,5,7,7],[2,6,6,8],[3,7,5,9],[3,8,4,4]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,16,10,17],[7,4,8,5],[19,3,20,4],[1,15,2,16],[10,18,11,17],[5,11,6,12],[12,6,13,7],[13,18,14,19],[14,2,15,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(14,1,-15,-2)(5,2,-6,-3)(19,4,-20,-5)(3,18,-4,-19)(7,10,-8,-11)(16,11,-17,-12)(12,17,-13,-18)(20,13,-9,-14)(6,15,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,-9)(-2,5,-20,-14)(-3,-19,-5)(-4,19)(-6,-16,-12,-18,3)(-7,-11,16)(-8,9,13,17,11)(-10,7,15,1)(-13,20,4,18)(-15,6,2)(-17,12)(8,10)
Multiloop annotated with half-edges
12^2_328 annotated with half-edges