$12^{2}_{106}$ - Minimal pinning sets
Pinning sets for 12^2_106 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_106 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, 9, 10}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 8, 9, 11}
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, 5, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,7,3],[0,2,7,7],[0,8,8,1],[1,9,6,1],[2,5,9,7],[2,6,3,3],[4,9,9,4],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,6,16,5],[10,13,11,14],[11,19,12,20],[1,7,2,6],[16,4,17,5],[17,9,18,10],[18,12,19,13],[7,3,8,2],[8,3,9,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,14,-6,-1)(16,1,-17,-2)(2,15,-3,-16)(3,12,-4,-13)(17,6,-18,-7)(18,9,-19,-10)(7,10,-8,-11)(13,4,-14,-5)(8,19,-9,-20)(11,20,-12,-15)
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,-3,-13,-5)(-2,-16)(-4,13)(-6,17,1)(-7,-11,-15,2,-17)(-8,-20,11)(-9,18,6,14,4,12,20)(-10,7,-18)(-12,3,15)(-14,5)(-19,8,10)(9,19)
Multiloop annotated with half-edges
12^2_106 annotated with half-edges