$12^{2}_{126}$ - Minimal pinning sets
Pinning sets for 12^2_126 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_126 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97043
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 5, 6, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 6, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,4],[0,5,3,0],[0,2,5,1],[1,6,7,1],[2,8,9,3],[4,9,7,7],[4,6,6,8],[5,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[16,7,1,8],[8,14,9,13],[6,15,7,16],[1,15,2,14],[9,12,10,13],[5,2,6,3],[11,20,12,17],[10,20,11,19],[3,19,4,18],[4,17,5,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,4,-14,-5)(6,9,-7,-10)(10,5,-11,-6)(11,14,-12,-15)(3,12,-4,-13)(2,15,-3,-16)(17,16,-18,-1)(18,7,-19,-8)(8,19,-9,-20)(1,20,-2,-17)
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)(-2,-16,17)(-3,-13,-5,10,-7,18,16)(-4,13)(-6,-10)(-8,-20,1,-18)(-9,6,-11,-15,2,20)(-12,3,15)(-14,11,5)(-19,8)(4,12,14)(7,9,19)
Multiloop annotated with half-edges
12^2_126 annotated with half-edges