$9^{1}_{10}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 4
- Total number of pinning sets: 32
- 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: 2.72966
- 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, 5, 6} |
4 |
[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 |
| 4 |
1 |
0 |
0 |
2.0 |
| 5 |
0 |
0 |
5 |
2.4 |
| 6 |
0 |
0 |
10 |
2.67 |
| 7 |
0 |
0 |
10 |
2.86 |
| 8 |
0 |
0 |
5 |
3.0 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
1 |
0 |
31 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 2, 2, 3, 4, 4, 4, 5]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,5,5,6],[0,6,6,0],[1,6,5,1],[1,4,2,2],[2,4,3,3]]
- PD code (use to draw this loop with SnapPy): [[5,14,6,1],[11,4,12,5],[13,8,14,9],[6,2,7,1],[3,10,4,11],[12,10,13,9],[7,2,8,3]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (8,1,-9,-2)(10,5,-11,-6)(6,9,-7,-10)(14,7,-1,-8)(2,11,-3,-12)(12,3,-13,-4)(4,13,-5,-14)
- 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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,-12,-4,-14,-8)(-3,12)(-5,10,-7,14)(-6,-10)(-9,6,-11,2)(-13,4)(1,7,9)(3,11,5,13)
Loop annotated with half-edges