$8^{1}_{1}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 4
- 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: 2.62679
- 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, 5, 7} |
4 |
[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 |
| 4 |
1 |
0 |
0 |
2.0 |
| 5 |
0 |
0 |
4 |
2.4 |
| 6 |
0 |
0 |
6 |
2.67 |
| 7 |
0 |
0 |
4 |
2.86 |
| 8 |
0 |
0 |
1 |
3.0 |
| Total |
1 |
0 |
15 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 2, 2, 3, 3, 5, 5]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,4],[0,5,5,1],[2,5,5,2],[3,4,4,3]]
- PD code (use to draw this loop with SnapPy): [[12,5,1,6],[6,11,7,12],[7,4,8,5],[1,10,2,11],[3,8,4,9],[9,2,10,3]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (4,1,-5,-2)(9,2,-10,-3)(12,5,-1,-6)(10,7,-11,-8)(3,8,-4,-9)(6,11,-7,-12)
- 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)
- Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,8,-11,6)(-2,9,-4)(-3,-9)(-5,12,-7,10,2)(-6,-12)(-8,3,-10)(1,5)(7,11)
Loop annotated with half-edges