$8^{2}_{3}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 4
- Total number of pinning sets: 43
- of which optimal: 5
- of which minimal: 5
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.83367
- on average over minimal pinning sets: 2.6
- on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{1, 3, 4, 6} |
4 |
[2, 2, 3, 3] |
2.50 |
| B (optimal) |
• |
{1, 3, 5, 8} |
4 |
[2, 2, 4, 4] |
3.00 |
| C (optimal) |
• |
{1, 3, 6, 7} |
4 |
[2, 2, 3, 3] |
2.50 |
| D (optimal) |
• |
{1, 2, 3, 4} |
4 |
[2, 2, 3, 3] |
2.50 |
| E (optimal) |
• |
{1, 2, 3, 7} |
4 |
[2, 2, 3, 3] |
2.50 |
Data for pinning sets in each cardinal
| Cardinality |
Optimal pinning sets |
Minimal suboptimal pinning sets |
Nonminimal pinning sets |
Averaged mean-degree |
| 4 |
5 |
0 |
0 |
2.6 |
| 5 |
0 |
0 |
16 |
2.8 |
| 6 |
0 |
0 |
15 |
2.89 |
| 7 |
0 |
0 |
6 |
2.95 |
| 8 |
0 |
0 |
1 |
3.0 |
| Total |
5 |
0 |
38 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,5,4,3],[0,2,1,0],[1,2,5,5],[1,4,4,2]]
- PD code (use to draw this multiloop with SnapPy): [[6,12,1,7],[7,11,8,10],[5,2,6,3],[11,1,12,2],[8,5,9,4],[9,3,10,4]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (7,6,-8,-1)(1,12,-2,-7)(9,4,-10,-5)(2,5,-3,-6)(8,11,-9,-12)(3,10,-4,-11)
- 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,-7)(-2,-6,7)(-3,-11,8,6)(-4,9,11)(-5,2,12,-9)(-8,-12,1)(-10,3,5)(4,10)
Multiloop annotated with half-edges