$9^{2}_{1}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 3
- Total number of pinning sets: 64
- 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: 2.79396
- 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, 3, 6} |
3 |
[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 |
| 3 |
1 |
0 |
0 |
2.0 |
| 4 |
0 |
0 |
6 |
2.42 |
| 5 |
0 |
0 |
15 |
2.67 |
| 6 |
0 |
0 |
20 |
2.83 |
| 7 |
0 |
0 |
15 |
2.95 |
| 8 |
0 |
0 |
6 |
3.04 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
1 |
0 |
63 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 5, 5]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,3],[0,3,4,4],[0,5,6,3],[0,2,1,0],[1,6,5,1],[2,4,6,6],[2,5,5,4]]
- PD code (use to draw this multiloop with SnapPy): [[6,14,1,7],[7,13,8,12],[5,2,6,3],[13,1,14,2],[8,11,9,12],[3,9,4,10],[10,4,11,5]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (3,6,-4,-1)(7,2,-8,-3)(10,5,-11,-6)(1,14,-2,-7)(13,8,-14,-9)(9,12,-10,-13)(4,11,-5,-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,-7,-3)(-2,7)(-4,-12,9,-14,1)(-5,10,12)(-6,3,-8,13,-10)(-9,-13)(-11,4,6)(2,14,8)(5,11)
Multiloop annotated with half-edges