$12^{3}_{68}$ - Minimal pinning sets
Pinning sets for 12^3_68 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_68 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 384
of which optimal: 1
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10338
on average over minimal pinning sets: 2.56667
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, 5, 6, 9}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {2, 3, 4, 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
4
1
0
0
2.5
5
0
2
8
2.72
6
0
0
40
2.89
7
0
0
86
3.02
8
0
0
110
3.12
9
0
0
86
3.2
10
0
0
40
3.26
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
2
381
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],[0,1,5,3],[0,2,6,7],[0,8,5,1],[1,4,9,2],[3,9,7,7],[3,6,6,8],[4,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[4,16,1,5],[5,10,6,11],[11,3,12,4],[12,15,13,16],[1,9,2,10],[6,2,7,3],[14,20,15,17],[13,20,14,19],[8,18,9,19],[7,18,8,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(12,3,-13,-4)(2,15,-3,-16)(11,16,-12,-5)(6,9,-7,-10)(1,10,-2,-11)(18,13,-19,-14)(14,19,-15,-20)(20,7,-17,-8)(8,17,-9,-18)
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,-16,11)(-3,12,16)(-4,5,-12)(-6,-10,1)(-7,20,-15,2,10)(-8,-18,-14,-20)(-9,6,4,-13,18)(-17,8)(-19,14)(3,15,19,13)(7,9,17)
Multiloop annotated with half-edges
12^3_68 annotated with half-edges