$9^{1}_{9}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 4
- Total number of pinning sets: 44
- of which optimal: 1
- of which minimal: 3
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.8232
- on average over minimal pinning sets: 2.41667
- 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, 4, 8} |
4 |
[2, 2, 2, 3] |
2.25 |
| a (minimal) |
• |
{1, 4, 5, 8, 9} |
5 |
[2, 2, 2, 3, 3] |
2.40 |
| b (minimal) |
• |
{1, 3, 4, 8, 9} |
5 |
[2, 2, 2, 3, 4] |
2.60 |
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 |
2 |
5 |
2.54 |
| 6 |
0 |
0 |
15 |
2.78 |
| 7 |
0 |
0 |
14 |
2.94 |
| 8 |
0 |
0 |
6 |
3.04 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
1 |
2 |
41 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 5]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,2,3],[0,4,4,2],[0,1,5,0],[0,5,6,6],[1,6,5,1],[2,4,6,3],[3,5,4,3]]
- PD code (use to draw this loop with SnapPy): [[5,14,6,1],[4,7,5,8],[13,6,14,7],[1,10,2,11],[8,3,9,4],[9,12,10,13],[2,12,3,11]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (6,1,-7,-2)(11,4,-12,-5)(2,5,-3,-6)(13,8,-14,-9)(9,14,-10,-1)(7,10,-8,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-3,-13,-9)(-2,-6)(-4,11,-8,13)(-5,2,-7,-11)(-10,7,1)(-12,3,5)(-14,9)(4,12)(8,10,14)
Loop annotated with half-edges