$12^{1}_{71}$ - Minimal pinning sets
Pinning sets for 12^1_71 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_71 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, 3, 6, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 4, 6, 7, 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, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,4,0],[0,5,3,0],[1,2,6,7],[1,8,5,5],[2,4,4,6],[3,5,9,7],[3,6,9,8],[4,7,9,9],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[19,4,20,5],[6,2,7,1],[7,18,8,19],[3,14,4,15],[2,14,3,13],[17,12,18,13],[8,12,9,11],[15,11,16,10],[16,9,17,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(13,6,-14,-7)(7,12,-8,-13)(8,5,-9,-6)(14,9,-15,-10)(10,19,-11,-20)(20,11,-1,-12)(4,15,-5,-16)(2,17,-3,-18)(18,3,-19,-4)
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,-5,8,12)(-2,-18,-4,-16)(-3,18)(-6,13,-8)(-7,-13)(-9,14,6)(-10,-20,-12,7,-14)(-11,20)(-15,4,-19,10)(-17,2)(1,11,19,3,17)(5,15,9)
Loop annotated with half-edges
12^1_71 annotated with half-edges