$9^{1}_{6}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 5
- Total number of pinning sets: 30
- of which optimal: 4
- of which minimal: 4
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.75164
- on average over minimal pinning sets: 2.3
- on average over optimal pinning sets: 2.3
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{2, 3, 4, 6, 8} |
5 |
[2, 2, 2, 2, 4] |
2.40 |
| B (optimal) |
• |
{2, 3, 5, 6, 8} |
5 |
[2, 2, 2, 2, 3] |
2.20 |
| C (optimal) |
• |
{2, 3, 6, 7, 8} |
5 |
[2, 2, 2, 2, 4] |
2.40 |
| D (optimal) |
• |
{1, 2, 3, 6, 8} |
5 |
[2, 2, 2, 2, 3] |
2.20 |
Data for pinning sets in each cardinal
| Cardinality |
Optimal pinning sets |
Minimal suboptimal pinning sets |
Nonminimal pinning sets |
Averaged mean-degree |
| 5 |
4 |
0 |
0 |
2.3 |
| 6 |
0 |
0 |
10 |
2.67 |
| 7 |
0 |
0 |
10 |
2.86 |
| 8 |
0 |
0 |
5 |
3.0 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
4 |
0 |
26 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 6]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,4,5,3],[0,2,6,6],[1,5,2,1],[2,4,6,6],[3,5,5,3]]
- PD code (use to draw this loop with SnapPy): [[14,5,1,6],[6,13,7,14],[4,11,5,12],[1,11,2,10],[12,7,13,8],[8,3,9,4],[2,9,3,10]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (5,14,-6,-1)(11,2,-12,-3)(9,4,-10,-5)(13,6,-14,-7)(7,12,-8,-13)(1,8,-2,-9)(3,10,-4,-11)
- 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,-9,-5)(-2,11,-4,9)(-3,-11)(-6,13,-8,1)(-7,-13)(-10,3,-12,7,-14,5)(2,8,12)(4,10)(6,14)
Loop annotated with half-edges