$8^{1}_{2}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 4
- Total number of pinning sets: 20
- 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: 2.72107
- on average over minimal pinning sets: 2.325
- on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{1, 2, 5, 8} |
4 |
[2, 2, 2, 3] |
2.25 |
| a (minimal) |
• |
{1, 2, 4, 5, 7} |
5 |
[2, 2, 2, 3, 3] |
2.40 |
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 |
2.25 |
| 5 |
0 |
1 |
4 |
2.52 |
| 6 |
0 |
0 |
8 |
2.75 |
| 7 |
0 |
0 |
5 |
2.91 |
| 8 |
0 |
0 |
1 |
3.0 |
| Total |
1 |
1 |
18 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 5]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,3],[0,2,5,0],[1,5,5,1],[2,4,4,3]]
- PD code (use to draw this loop with SnapPy): [[12,7,1,8],[8,4,9,3],[11,2,12,3],[6,1,7,2],[4,10,5,9],[5,10,6,11]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (9,12,-10,-1)(1,6,-2,-7)(10,5,-11,-6)(7,2,-8,-3)(3,8,-4,-9)(4,11,-5,-12)
- 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)
- Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-7,-3,-9)(-2,7)(-4,-12,9)(-5,10,12)(-6,1,-10)(-8,3)(-11,4,8,2,6)(5,11)
Loop annotated with half-edges