$9^{1}_{5}$ - 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: 2.91875
- on average over minimal pinning sets: 2.61
- on average over optimal pinning sets: 2.625
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{1, 3, 4, 6} |
4 |
[2, 2, 3, 4] |
2.75 |
| B (optimal) |
• |
{1, 2, 4, 8} |
4 |
[2, 2, 3, 3] |
2.50 |
| a (minimal) |
• |
{1, 3, 4, 7, 8} |
5 |
[2, 2, 3, 3, 3] |
2.60 |
| b (minimal) |
• |
{1, 3, 4, 5, 7} |
5 |
[2, 2, 3, 3, 3] |
2.60 |
| c (minimal) |
• |
{1, 2, 3, 4, 5} |
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 |
| 4 |
2 |
0 |
0 |
2.62 |
| 5 |
0 |
3 |
10 |
2.75 |
| 6 |
0 |
0 |
24 |
2.91 |
| 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, 3, 4, 5]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,6,4,3],[0,2,1,0],[1,2,6,5],[1,4,6,6],[2,5,5,4]]
- PD code (use to draw this loop with SnapPy): [[14,7,1,8],[8,6,9,5],[13,2,14,3],[6,1,7,2],[9,13,10,12],[4,11,5,12],[3,11,4,10]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (2,5,-3,-6)(11,4,-12,-5)(7,14,-8,-1)(1,8,-2,-9)(9,6,-10,-7)(3,12,-4,-13)(10,13,-11,-14)
- 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,-7)(-2,-6,9)(-3,-13,10,6)(-4,11,13)(-5,2,8,14,-11)(-8,1)(-10,-14,7)(-12,3,5)(4,12)
Loop annotated with half-edges