$7^{2}_{1}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 4
- Total number of pinning sets: 14
- 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: 2.70408
- 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, 2, 4, 7} |
4 |
[2, 2, 3, 3] |
2.50 |
| B (optimal) |
• |
{1, 2, 5, 6} |
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 |
6 |
2.67 |
| 6 |
0 |
0 |
5 |
2.8 |
| 7 |
0 |
0 |
1 |
2.86 |
| Total |
2 |
0 |
12 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 2, 3, 3, 3, 3, 4]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,4,3],[0,2,4,4],[1,3,3,2]]
- PD code (use to draw this multiloop with SnapPy): [[4,10,1,5],[5,3,6,4],[6,9,7,10],[1,7,2,8],[8,2,9,3]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (5,4,-6,-1)(8,3,-9,-4)(6,9,-7,-10)(1,10,-2,-5)(2,7,-3,-8)
- Edge permutation $\epsilon=$ (-1,1)(-2,2)(-3,3)(-4,4)(-5,5)(-6,6)(-7,7)(-8,8)(-9,9)(-10,10)
- Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-5)(-2,-8,-4,5)(-3,8)(-6,-10,1)(-7,2,10)(-9,6,4)(3,7,9)
Multiloop annotated with half-edges