$12^{1}_{45}$ - Minimal pinning sets
Pinning sets for 12^1_45 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_45 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, 2, 3, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 6, 9}
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: [[1,2,3,3],[0,4,5,6],[0,7,7,8],[0,9,4,0],[1,3,5,5],[1,4,4,6],[1,5,9,7],[2,6,8,2],[2,7,9,9],[3,8,8,6]]
PD code (use to draw this loop with SnapPy): [[13,20,14,1],[12,9,13,10],[19,4,20,5],[14,2,15,1],[10,15,11,16],[16,11,17,12],[17,8,18,9],[5,18,6,19],[6,3,7,4],[2,7,3,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(17,2,-18,-3)(20,5,-1,-6)(15,6,-16,-7)(10,7,-11,-8)(14,9,-15,-10)(8,13,-9,-14)(3,16,-4,-17)(11,18,-12,-19)(19,12,-20,-13)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,16,6)(-2,17,-4)(-3,-17)(-5,20,12,18,2)(-6,15,9,13,-20)(-7,10,-15)(-8,-14,-10)(-9,14)(-11,-19,-13,8)(-12,19)(-16,3,-18,11,7)(1,5)
Loop annotated with half-edges
12^1_45 annotated with half-edges