$12^{3}_{3}$ - Minimal pinning sets
Pinning sets for 12^3_3 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_3 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 318
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: 3.03847
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, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 6, 9, 10}
5
[2, 2, 2, 3, 5]
2.80
C (optimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 3, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
E (optimal)
• {1, 2, 6, 7, 9}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 2, 6, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
G (optimal)
• {1, 2, 6, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
H (optimal)
• {1, 2, 5, 6, 9}
5
[2, 2, 2, 3, 5]
2.80
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
34
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
8
0
310
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: [[1,2,3,3],[0,4,4,5],[0,5,6,3],[0,2,4,0],[1,3,6,1],[1,7,8,2],[2,9,7,4],[5,6,9,8],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[10,14,1,11],[11,15,12,20],[9,2,10,3],[13,1,14,2],[15,13,16,12],[19,3,20,4],[8,16,9,17],[4,8,5,7],[18,6,19,7],[17,6,18,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,10,-8,-1)(12,1,-13,-2)(3,16,-4,-11)(11,4,-12,-5)(5,2,-6,-3)(20,9,-17,-10)(19,14,-20,-15)(8,17,-9,-18)(13,18,-14,-19)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,4,16,-7)(-2,5,-12)(-3,-11,-5)(-4,11)(-6,-16,3)(-8,-18,13,1)(-9,20,14,18)(-10,7,15,-20)(-13,-19,-15,6,2)(-14,19)(-17,8,10)(9,17)
Multiloop annotated with half-edges
12^3_3 annotated with half-edges