$9^{2}_{6}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 5
- Total number of pinning sets: 28
- of which optimal: 3
- of which minimal: 3
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.75533
- on average over minimal pinning sets: 2.26667
- on average over optimal pinning sets: 2.26667
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{1, 2, 3, 5, 6} |
5 |
[2, 2, 2, 2, 3] |
2.20 |
| B (optimal) |
• |
{1, 2, 3, 6, 9} |
5 |
[2, 2, 2, 2, 4] |
2.40 |
| C (optimal) |
• |
{1, 2, 3, 6, 8} |
5 |
[2, 2, 2, 2, 3] |
2.20 |
Data for pinning sets in each cardinal
| Cardinality |
Optimal pinning sets |
Minimal suboptimal pinning sets |
Nonminimal pinning sets |
Averaged mean-degree |
| 5 |
3 |
0 |
0 |
2.27 |
| 6 |
0 |
0 |
9 |
2.63 |
| 7 |
0 |
0 |
10 |
2.86 |
| 8 |
0 |
0 |
5 |
3.0 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
3 |
0 |
25 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 5, 5]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,5],[0,6,4,0],[1,3,6,1],[2,6,6,2],[3,5,5,4]]
- PD code (use to draw this multiloop with SnapPy): [[10,14,1,11],[11,6,12,5],[9,4,10,5],[13,1,14,2],[6,13,7,12],[3,8,4,9],[2,8,3,7]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (7,2,-8,-3)(3,14,-4,-11)(5,10,-6,-1)(1,6,-2,-7)(12,9,-13,-10)(11,4,-12,-5)(8,13,-9,-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,-7,-3,-11,-5)(-2,7)(-4,11)(-6,1)(-8,-14,3)(-9,12,4,14)(-10,5,-12)(-13,8,2,6,10)(9,13)
Multiloop annotated with half-edges