$12^{3}_{4}$ - Minimal pinning sets
Pinning sets for 12^3_4 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_4 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 315
of which optimal: 6
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04074
on average over minimal pinning sets: 2.60417
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)
• {2, 4, 6, 8, 12}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {2, 4, 5, 8, 12}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {2, 4, 5, 6, 12}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {2, 4, 6, 7, 12}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {2, 3, 4, 6, 12}
5
[2, 2, 2, 3, 4]
2.60
F (optimal)
• {2, 4, 6, 9, 12}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {2, 4, 6, 10, 11, 12}
6
[2, 2, 2, 3, 4, 6]
3.17
b (minimal)
• {1, 2, 4, 6, 11, 12}
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
6
0
0
2.5
6
0
2
31
2.77
7
0
0
71
2.94
8
0
0
90
3.06
9
0
0
71
3.16
10
0
0
34
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
6
2
307
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,7],[0,7,8,9],[0,9,1,1],[1,9,9,2],[2,8,8,7],[2,6,8,3],[3,7,6,6],[3,5,5,4]]
PD code (use to draw this multiloop with SnapPy): [[3,8,4,1],[2,12,3,9],[7,20,8,13],[4,16,5,15],[1,10,2,9],[11,13,12,14],[6,17,7,18],[19,16,20,17],[5,19,6,18],[14,10,15,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,6,-2,-7)(8,9,-1,-10)(10,7,-11,-8)(20,11,-13,-12)(13,2,-14,-3)(17,14,-18,-15)(4,15,-5,-16)(16,3,-17,-4)(5,18,-6,-19)(12,19,-9,-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,-7,10)(-2,13,11,7)(-3,16,-5,-19,12,-13)(-4,-16)(-6,1,9,19)(-8,-10)(-9,8,-11,20)(-12,-20)(-14,17,3)(-15,4,-17)(-18,5,15)(2,6,18,14)
Multiloop annotated with half-edges
12^3_4 annotated with half-edges