$9^{2}_{2}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 5
- 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: 2.75484
- 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, 4, 5, 6} |
5 |
[2, 2, 2, 2, 3] |
2.20 |
| a (minimal) |
• |
{1, 3, 4, 5, 6, 8} |
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 |
4 |
2.5 |
| 7 |
0 |
0 |
8 |
2.79 |
| 8 |
0 |
0 |
5 |
3.0 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
1 |
1 |
18 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 5, 6]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,1,2,2],[0,3,4,0],[0,5,5,0],[1,6,6,4],[1,3,6,5],[2,4,6,2],[3,5,4,3]]
- PD code (use to draw this multiloop with SnapPy): [[6,14,1,7],[7,5,8,6],[13,1,14,2],[10,4,11,5],[8,11,9,12],[2,12,3,13],[3,9,4,10]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (8,1,-9,-2)(13,2,-14,-3)(11,4,-12,-5)(6,7,-1,-8)(14,9,-7,-10)(5,10,-6,-11)(3,12,-4,-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)
- Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,13,-4,11,-6,-8)(-3,-13)(-5,-11)(-7,6,10)(-9,14,2)(-10,5,-12,3,-14)(1,7,9)(4,12)
Multiloop annotated with half-edges