$12^{4}_{12}$ - Minimal pinning sets
Pinning sets for 12^4_12 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_12 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 412
of which optimal: 2
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04651
on average over minimal pinning sets: 2.7
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, 3, 7, 11}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 5, 7, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 4, 7, 10, 11}
6
[2, 2, 2, 4, 4, 4]
3.00
b (minimal)
• {1, 2, 4, 7, 9, 11}
6
[2, 2, 2, 4, 4, 4]
3.00
c (minimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 2, 4, 4, 4]
3.00
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
0
15
2.59
6
0
3
49
2.82
7
0
0
100
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
3
407
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: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,7],[0,7,5,8],[0,8,8,1],[1,8,3,9],[1,9,9,2],[2,9,3,2],[3,5,4,4],[5,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[6,10,1,7],[7,11,8,16],[5,20,6,17],[9,3,10,4],[1,12,2,11],[8,13,9,14],[15,17,16,18],[19,4,20,5],[2,12,3,13],[14,19,15,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-13,-2)(17,2,-18,-3)(4,13,-5,-14)(11,14,-12,-15)(12,5,-7,-6)(6,7,-1,-8)(15,8,-16,-9)(20,9,-17,-10)(3,18,-4,-19)(10,19,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,8)(-2,17,9,-16)(-3,-19,10,-17)(-4,-14,11,19)(-5,12,14)(-6,-8,15,-12)(-7,6)(-9,20,-11,-15)(-10,-20)(-13,4,18,2)(-18,3)(1,7,5,13)
Multiloop annotated with half-edges
12^4_12 annotated with half-edges