$6^{1}_{1}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 4
- Total number of pinning sets: 7
- of which optimal: 2
- of which minimal: 2
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.58095
- on average over minimal pinning sets: 2.5
- 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, 3, 5, 6} |
4 |
[2, 2, 3, 3] |
2.50 |
| B (optimal) |
• |
{1, 2, 4, 5} |
4 |
[2, 2, 3, 3] |
2.50 |
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 |
0 |
4 |
2.6 |
| 6 |
0 |
0 |
1 |
2.67 |
| Total |
2 |
0 |
5 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 3, 3, 3, 3]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,3,3],[0,2,2,1]]
- PD code (use to draw this loop with SnapPy): [[8,3,1,4],[4,7,5,8],[5,2,6,3],[1,6,2,7]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (3,8,-4,-1)(6,1,-7,-2)(7,4,-8,-5)(2,5,-3,-6)
- Edge permutation $\epsilon=$ (-1,1)(-2,2)(-3,3)(-4,4)(-5,5)(-6,6)(-7,7)(-8,8)
- Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-3)(-2,-6)(-4,7,1)(-5,2,-7)(-8,3,5)(4,8)
Loop annotated with half-edges