$12^{1}_{86}$ - Minimal pinning sets
Pinning sets for 12^1_86 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_86 Pinning data
Pinning number of this loop: 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, 3, 10, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 7, 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 loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,4,0],[0,5,5,6],[0,6,7,1],[1,8,5,5],[2,4,4,2],[2,9,7,3],[3,6,9,8],[4,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[19,10,20,11],[12,8,13,7],[1,18,2,19],[9,14,10,15],[8,14,9,13],[17,6,18,7],[2,6,3,5],[15,5,16,4],[16,3,17,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,11,-1,-12)(1,18,-2,-19)(13,2,-14,-3)(3,8,-4,-9)(15,6,-16,-7)(9,4,-10,-5)(5,10,-6,-11)(17,14,-18,-15)(7,16,-8,-17)(12,19,-13,-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,-19,12)(-2,13,19)(-3,-9,-5,-11,20,-13)(-4,9)(-6,15,-18,1,11)(-7,-17,-15)(-8,3,-14,17)(-10,5)(-12,-20)(-16,7)(2,18,14)(4,8,16,6,10)
Loop annotated with half-edges
12^1_86 annotated with half-edges