$12^{1}_{228}$ - Minimal pinning sets
Pinning sets for 12^1_228 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_228 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
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
B (optimal)
• {1, 2, 3, 6, 7}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
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: [[1,1,2,3],[0,4,4,0],[0,4,5,6],[0,6,7,8],[1,8,2,1],[2,9,6,6],[2,5,5,3],[3,9,9,8],[3,7,9,4],[5,8,7,7]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[19,12,20,13],[14,18,15,17],[1,6,2,7],[11,18,12,19],[15,4,16,5],[5,16,6,17],[2,9,3,10],[7,10,8,11],[8,3,9,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(13,2,-14,-3)(19,4,-20,-5)(20,7,-1,-8)(5,8,-6,-9)(17,10,-18,-11)(3,14,-4,-15)(15,12,-16,-13)(9,16,-10,-17)(11,18,-12,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,8)(-2,13,-16,9,-6)(-3,-15,-13)(-4,19,-12,15)(-5,-9,-17,-11,-19)(-7,20,4,14,2)(-8,5,-20)(-10,17)(-14,3)(-18,11)(1,7)(10,16,12,18)
Loop annotated with half-edges
12^1_228 annotated with half-edges