$12^{1}_{359}$ - Minimal pinning sets
Pinning sets for 12^1_359 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_359 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 256
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96564
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 6, 9}
4
[2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,7,0],[0,8,8,1],[1,8,9,9],[1,9,6,6],[2,5,5,7],[2,6,9,8],[3,7,4,3],[4,7,5,4]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[8,13,9,14],[19,10,20,11],[1,15,2,14],[16,7,17,8],[12,5,13,6],[11,5,12,4],[18,3,19,4],[15,3,16,2],[6,17,7,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,20,-16,-1)(1,8,-2,-9)(9,2,-10,-3)(13,4,-14,-5)(16,7,-17,-8)(5,10,-6,-11)(11,18,-12,-19)(3,14,-4,-15)(6,17,-7,-18)(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,-9,-3,-15)(-2,9)(-4,13,-20,15)(-5,-11,-19,-13)(-6,-18,11)(-7,16,20,12,18)(-8,1,-16)(-10,5,-14,3)(-12,19)(-17,6,10,2,8)(4,14)(7,17)
Loop annotated with half-edges
12^1_359 annotated with half-edges