$12^{3}_{72}$ - Minimal pinning sets
Pinning sets for 12^3_72 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_72 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 368
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10917
on average over minimal pinning sets: 2.66667
on average over optimal pinning sets: 2.66667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 6, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {2, 3, 6, 9, 10}
5
[2, 2, 3, 3, 4]
2.80
C (optimal)
• {2, 3, 6, 9, 11}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {1, 4, 5, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
E (optimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
F (optimal)
• {1, 2, 3, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
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
2.67
6
0
0
34
2.87
7
0
0
82
3.02
8
0
0
109
3.12
9
0
0
86
3.2
10
0
0
40
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
6
0
362
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,7,8,8],[0,9,5,1],[1,4,9,2],[2,9,7,7],[3,6,6,8],[3,7,9,3],[4,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[4,10,1,5],[5,11,6,20],[3,19,4,20],[9,16,10,17],[1,12,2,11],[6,2,7,3],[13,18,14,19],[17,14,18,15],[15,8,16,9],[12,8,13,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(6,13,-7,-14)(16,7,-17,-8)(2,9,-3,-10)(18,15,-19,-16)(8,17,-9,-18)(14,19,-15,-20)(1,20,-2,-11)(11,10,-12,-5)(12,3,-13,-4)
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,-11,-5)(-2,-10,11)(-3,12,10)(-4,5,-12)(-6,-14,-20,1)(-7,16,-19,14)(-8,-18,-16)(-9,2,20,-15,18)(-13,6,4)(-17,8)(3,9,17,7,13)(15,19)
Multiloop annotated with half-edges
12^3_72 annotated with half-edges