$12^{2}_{331}$ - Minimal pinning sets
Pinning sets for 12^2_331 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_331 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 384
of which optimal: 8
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.06002
on average over minimal pinning sets: 2.55
on average over optimal pinning sets: 2.55
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 9, 12}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 4, 8, 12}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 2, 4, 10, 12}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 2, 4, 6, 10}
5
[2, 2, 2, 4, 4]
2.80
E (optimal)
• {1, 2, 4, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
G (optimal)
• {1, 2, 4, 7, 12}
5
[2, 2, 2, 3, 3]
2.40
H (optimal)
• {1, 2, 4, 7, 10}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
8
0
0
2.55
6
0
0
42
2.8
7
0
0
93
2.97
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
8
0
376
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,3],[0,2,5,8],[0,8,8,1],[1,8,3,7],[1,9,9,2],[2,9,9,5],[3,5,4,4],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,11,18,12],[15,4,16,5],[19,3,20,4],[1,10,2,11],[18,9,19,8],[12,6,13,5],[7,14,8,15],[9,2,10,3],[6,14,7,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,16,-8,-1)(18,1,-19,-2)(13,2,-14,-3)(3,10,-4,-11)(6,17,-7,-18)(15,8,-16,-9)(20,9,-17,-10)(11,4,-12,-5)(5,12,-6,-13)(14,19,-15,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-7)(-2,13,-6,-18)(-3,-11,-5,-13)(-4,11)(-8,15,19,1)(-9,20,-15)(-10,3,-14,-20)(-12,5)(-16,7,17,9)(-17,6,12,4,10)(-19,14,2)(8,16)
Multiloop annotated with half-edges
12^2_331 annotated with half-edges