$9^{2}_{3}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 4
- Total number of pinning sets: 40
- of which optimal: 1
- of which minimal: 2
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.81611
- on average over minimal pinning sets: 2.325
- on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{1, 2, 5, 6} |
4 |
[2, 2, 2, 3] |
2.25 |
| a (minimal) |
• |
{1, 2, 3, 6, 7} |
5 |
[2, 2, 2, 3, 3] |
2.40 |
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 |
2.25 |
| 5 |
0 |
1 |
5 |
2.53 |
| 6 |
0 |
0 |
13 |
2.76 |
| 7 |
0 |
0 |
13 |
2.92 |
| 8 |
0 |
0 |
6 |
3.04 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
1 |
1 |
38 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 5]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,3],[0,2,6,6],[0,6,6,5],[1,4,2,1],[3,4,4,3]]
- PD code (use to draw this multiloop with SnapPy): [[8,14,1,9],[9,6,10,5],[7,4,8,5],[13,3,14,4],[1,12,2,11],[6,11,7,10],[2,12,3,13]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (10,1,-11,-2)(5,2,-6,-3)(3,12,-4,-13)(13,4,-14,-5)(14,7,-9,-8)(8,9,-1,-10)(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)(-13,13)(-14,14)
- Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,5,-14,-8,-10)(-3,-13,-5)(-4,13)(-6,-12,3)(-7,14,4,12)(-9,8)(-11,6,2)(1,9,7,11)
Multiloop annotated with half-edges