$12^{3}_{76}$ - Minimal pinning sets
Pinning sets for 12^3_76 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_76 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 416
of which optimal: 2
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04592
on average over minimal pinning sets: 2.43333
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 7}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 2, 3, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 2, 4, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.25
5
0
1
15
2.6
6
0
0
54
2.83
7
0
0
101
2.98
8
0
0
115
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
2
1
413
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,7],[0,7,8,8],[0,8,6,4],[0,3,5,5],[1,4,4,6],[1,5,3,9],[1,9,9,2],[2,9,3,2],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[12,16,1,13],[13,3,14,4],[11,20,12,17],[15,9,16,10],[1,9,2,8],[2,7,3,8],[14,7,15,6],[4,18,5,17],[19,10,20,11],[5,18,6,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (1,20,-2,-17)(9,2,-10,-3)(3,8,-4,-9)(17,4,-18,-5)(14,5,-15,-6)(19,10,-20,-11)(7,18,-8,-19)(6,15,-7,-16)(16,11,-13,-12)(12,13,-1,-14)
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,-17,-5,14)(-2,9,-4,17)(-3,-9)(-6,-16,-12,-14)(-7,-19,-11,16)(-8,3,-10,19)(-13,12)(-15,6)(-18,7,15,5)(-20,1,13,11)(2,20,10)(4,8,18)
Multiloop annotated with half-edges
12^3_76 annotated with half-edges