$12^{2}_{329}$ - Minimal pinning sets
Pinning sets for 12^2_329 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_329 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 388
of which optimal: 8
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05998
on average over minimal pinning sets: 2.60667
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)
• {2, 3, 4, 6, 7}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {2, 3, 4, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {2, 3, 6, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {2, 3, 5, 7, 9}
5
[2, 2, 2, 3, 5]
2.80
E (optimal)
• {1, 2, 3, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {2, 3, 7, 10, 11}
5
[2, 2, 2, 4, 4]
2.80
G (optimal)
• {2, 3, 6, 7, 11}
5
[2, 2, 2, 3, 4]
2.60
H (optimal)
• {2, 3, 7, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {2, 3, 4, 7, 8, 10}
6
[2, 2, 2, 3, 4, 5]
3.00
b (minimal)
• {1, 2, 3, 4, 7, 10}
6
[2, 2, 2, 3, 3, 4]
2.67
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
2
41
2.8
7
0
0
95
2.98
8
0
0
114
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
2
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: [[1,2,2,3],[0,3,4,5],[0,6,7,0],[0,8,8,1],[1,8,7,9],[1,9,9,6],[2,5,9,7],[2,6,4,8],[3,7,4,3],[4,6,5,5]]
PD code (use to draw this multiloop with SnapPy): [[14,9,1,10],[10,15,11,20],[8,13,9,14],[1,16,2,15],[11,17,12,18],[19,5,20,6],[7,4,8,5],[12,3,13,4],[16,3,17,2],[18,7,19,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,14,-10,-1)(4,1,-5,-2)(15,2,-16,-3)(3,20,-4,-15)(11,6,-12,-7)(5,10,-6,-11)(13,18,-14,-19)(8,19,-9,-20)(16,7,-17,-8)(17,12,-18,-13)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,20,-9)(-2,15,-4)(-3,-15)(-5,-11,-7,16,2)(-6,11)(-8,-20,3,-16)(-10,5,1)(-12,17,7)(-13,-19,8,-17)(-14,9,19)(-18,13)(6,10,14,18,12)
Multiloop annotated with half-edges
12^2_329 annotated with half-edges