$9^{1}_{7}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 4
- Total number of pinning sets: 66
- of which optimal: 2
- of which minimal: 5
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.9094
- on average over minimal pinning sets: 2.68
- on average over optimal pinning sets: 2.5
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, 3, 3] |
2.50 |
| B (optimal) |
• |
{1, 2, 3, 6} |
4 |
[2, 2, 3, 3] |
2.50 |
| a (minimal) |
• |
{1, 2, 3, 7, 8} |
5 |
[2, 2, 3, 3, 4] |
2.80 |
| b (minimal) |
• |
{1, 2, 3, 4, 8} |
5 |
[2, 2, 3, 3, 4] |
2.80 |
| c (minimal) |
• |
{1, 2, 5, 6, 9} |
5 |
[2, 2, 3, 3, 4] |
2.80 |
Data for pinning sets in each cardinal
| Cardinality |
Optimal pinning sets |
Minimal suboptimal pinning sets |
Nonminimal pinning sets |
Averaged mean-degree |
| 4 |
2 |
0 |
0 |
2.5 |
| 5 |
0 |
3 |
10 |
2.74 |
| 6 |
0 |
0 |
24 |
2.9 |
| 7 |
0 |
0 |
19 |
3.01 |
| 8 |
0 |
0 |
7 |
3.07 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
2 |
3 |
61 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,6,4],[0,3,6,5],[1,4,2,1],[2,4,3,3]]
- PD code (use to draw this loop with SnapPy): [[7,14,8,1],[6,11,7,12],[13,10,14,11],[8,3,9,4],[1,4,2,5],[12,5,13,6],[2,9,3,10]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (6,1,-7,-2)(13,2,-14,-3)(10,3,-11,-4)(14,7,-1,-8)(11,8,-12,-9)(4,9,-5,-10)(5,12,-6,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,12,8)(-2,13,-6)(-3,10,-5,-13)(-4,-10)(-7,14,2)(-8,11,3,-14)(-9,4,-11)(-12,5,9)(1,7)
Loop annotated with half-edges