$8^{2}_{4}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 4
- Total number of pinning sets: 26
- of which optimal: 2
- of which minimal: 3
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.72582
- on average over minimal pinning sets: 2.43333
- 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, 3, 7, 8} |
4 |
[2, 2, 2, 3] |
2.25 |
| B (optimal) |
• |
{1, 3, 4, 7} |
4 |
[2, 2, 2, 3] |
2.25 |
| a (minimal) |
• |
{1, 2, 3, 5, 7} |
5 |
[2, 2, 2, 4, 4] |
2.80 |
Data for pinning sets in each cardinal
| Cardinality |
Optimal pinning sets |
Minimal suboptimal pinning sets |
Nonminimal pinning sets |
Averaged mean-degree |
| 4 |
2 |
0 |
0 |
2.25 |
| 5 |
0 |
1 |
7 |
2.6 |
| 6 |
0 |
0 |
10 |
2.8 |
| 7 |
0 |
0 |
5 |
2.91 |
| 8 |
0 |
0 |
1 |
3.0 |
| Total |
2 |
1 |
23 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 2, 2, 3, 3, 4, 4, 4]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,5,5,3],[0,2,4,0],[1,3,5,1],[1,4,2,2]]
- PD code (use to draw this multiloop with SnapPy): [[8,12,1,9],[9,5,10,6],[7,2,8,3],[11,1,12,2],[4,10,5,11],[6,4,7,3]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (8,3,-1,-4)(9,2,-10,-3)(11,6,-12,-7)(4,7,-5,-8)(1,10,-2,-11)(5,12,-6,-9)
- 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)
- Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-11,-7,4)(-2,9,-6,11)(-3,8,-5,-9)(-4,-8)(-10,1,3)(-12,5,7)(2,10)(6,12)
Multiloop annotated with half-edges