$9^{1}_{8}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 5
- Total number of pinning sets: 48
- of which optimal: 6
- of which minimal: 6
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.91184
- on average over minimal pinning sets: 2.63333
- on average over optimal pinning sets: 2.63333
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{1, 3, 4, 6, 8} |
5 |
[2, 2, 3, 3, 3] |
2.60 |
| B (optimal) |
• |
{1, 3, 5, 6, 8} |
5 |
[2, 2, 3, 3, 3] |
2.60 |
| C (optimal) |
• |
{1, 2, 5, 6, 8} |
5 |
[2, 2, 3, 3, 4] |
2.80 |
| D (optimal) |
• |
{1, 4, 5, 8, 9} |
5 |
[2, 2, 3, 3, 3] |
2.60 |
| E (optimal) |
• |
{1, 5, 6, 8, 9} |
5 |
[2, 2, 3, 3, 3] |
2.60 |
| F (optimal) |
• |
{1, 3, 4, 8, 9} |
5 |
[2, 2, 3, 3, 3] |
2.60 |
Data for pinning sets in each cardinal
| Cardinality |
Optimal pinning sets |
Minimal suboptimal pinning sets |
Nonminimal pinning sets |
Averaged mean-degree |
| 5 |
6 |
0 |
0 |
2.63 |
| 6 |
0 |
0 |
17 |
2.85 |
| 7 |
0 |
0 |
17 |
2.99 |
| 8 |
0 |
0 |
7 |
3.07 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
6 |
0 |
42 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 5]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,3],[0,2,6,4],[0,3,6,6],[1,6,2,1],[3,5,4,4]]
- PD code (use to draw this loop with SnapPy): [[5,14,6,1],[4,11,5,12],[13,10,14,11],[6,10,7,9],[1,9,2,8],[12,3,13,4],[7,3,8,2]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (14,5,-1,-6)(10,1,-11,-2)(6,13,-7,-14)(7,4,-8,-5)(11,8,-12,-9)(2,9,-3,-10)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-3,-13,6)(-2,-10)(-4,7,13)(-5,14,-7)(-6,-14)(-8,11,1,5)(-9,2,-11)(-12,3,9)(4,12,8)
Loop annotated with half-edges