$12^{2}_{306}$ - Minimal pinning sets
Pinning sets for 12^2_306 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_306 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 416
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10935
on average over minimal pinning sets: 2.625
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)
• {1, 4, 7, 12}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 3, 5, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {1, 4, 5, 7, 11}
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
3
8
2.73
6
0
0
45
2.9
7
0
0
96
3.04
8
0
0
120
3.14
9
0
0
91
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
3
412
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 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],[0,6,6,7],[0,7,5,1],[1,4,8,2],[2,9,3,3],[3,9,8,4],[5,7,9,9],[6,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,16,6,17],[17,3,18,4],[10,19,11,20],[1,15,2,16],[6,2,7,3],[18,9,19,10],[11,14,12,15],[7,12,8,13],[13,8,14,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(16,1,-17,-2)(12,7,-13,-8)(17,8,-18,-9)(6,11,-7,-12)(10,13,-11,-14)(3,14,-4,-15)(20,15,-5,-16)(9,18,-10,-19)(2,19,-3,-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,-5)(-2,-20,-16)(-3,-15,20)(-4,5,15)(-6,-12,-8,17,1)(-7,12)(-9,-19,2,-17)(-10,-14,3,19)(-11,6,4,14)(-13,10,18,8)(-18,9)(7,11,13)
Multiloop annotated with half-edges
12^2_306 annotated with half-edges