$12^{1}_{122}$ - Minimal pinning sets
Pinning sets for 12^1_122 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_122 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, 5, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 4, 6, 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, 3, 4, 6, 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],[0,8,9,1],[1,9,9,1],[2,8,8,7],[2,6,3,3],[4,6,6,9],[4,8,5,5]]
PD code (use to draw this loop with SnapPy): [[5,20,6,1],[11,4,12,5],[16,19,17,20],[6,17,7,18],[1,10,2,11],[3,12,4,13],[15,8,16,9],[18,7,19,8],[9,14,10,15],[2,14,3,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(15,4,-16,-5)(12,5,-13,-6)(18,7,-19,-8)(20,9,-1,-10)(10,19,-11,-20)(6,11,-7,-12)(3,14,-4,-15)(13,16,-14,-17)(2,17,-3,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-19,10)(-2,-18,-8)(-3,-15,-5,12,-7,18)(-4,15)(-6,-12)(-9,20,-11,6,-13,-17,2)(-10,-20)(-14,3,17)(-16,13,5)(1,9)(4,14,16)(7,11,19)
Loop annotated with half-edges
12^1_122 annotated with half-edges