$9^{1}_{3}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 5
- Total number of pinning sets: 16
- of which optimal: 1
- of which minimal: 1
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.67584
- on average over minimal pinning sets: 2.0
- on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{1, 2, 3, 6, 8} |
5 |
[2, 2, 2, 2, 2] |
2.00 |
Data for pinning sets in each cardinal
| Cardinality |
Optimal pinning sets |
Minimal suboptimal pinning sets |
Nonminimal pinning sets |
Averaged mean-degree |
| 5 |
1 |
0 |
0 |
2.0 |
| 6 |
0 |
0 |
4 |
2.42 |
| 7 |
0 |
0 |
6 |
2.71 |
| 8 |
0 |
0 |
4 |
2.94 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
1 |
0 |
15 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 2, 2, 2, 4, 4, 5, 5]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,1,2,2],[0,3,3,0],[0,4,5,0],[1,5,4,1],[2,3,6,6],[2,6,6,3],[4,5,5,4]]
- PD code (use to draw this loop with SnapPy): [[7,14,8,1],[13,6,14,7],[8,2,9,1],[5,12,6,13],[2,12,3,11],[9,4,10,5],[3,10,4,11]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (5,14,-6,-1)(9,2,-10,-3)(13,6,-14,-7)(7,12,-8,-13)(1,8,-2,-9)(3,10,-4,-11)(11,4,-12,-5)
- 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,-3,-11,-5)(-2,9)(-4,11)(-6,13,-8,1)(-7,-13)(-10,3)(-12,7,-14,5)(2,8,12,4,10)(6,14)
Loop annotated with half-edges