$9^{1}_{2}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 5
- 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: 2.67584
- 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, 3, 4, 8} |
5 |
[2, 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 |
| 5 |
1 |
0 |
0 |
2.0 |
| 6 |
0 |
0 |
4 |
2.42 |
| 7 |
0 |
0 |
6 |
2.71 |
| 8 |
0 |
0 |
4 |
2.94 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
1 |
0 |
15 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 6, 6]
- 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],[2,6,6,2],[3,6,6,3],[4,5,5,4]]
- PD code (use to draw this loop with SnapPy): [[7,14,8,1],[13,6,14,7],[8,6,9,5],[1,12,2,13],[9,4,10,5],[11,2,12,3],[3,10,4,11]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (9,14,-10,-1)(7,2,-8,-3)(3,6,-4,-7)(11,4,-12,-5)(1,8,-2,-9)(13,10,-14,-11)(5,12,-6,-13)
- 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,-9)(-2,7,-4,11,-14,9)(-3,-7)(-5,-13,-11)(-6,3,-8,1,-10,13)(-12,5)(2,8)(4,6,12)(10,14)
Loop annotated with half-edges