$9^{3}_{1}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 4
- Total number of pinning sets: 56
- of which optimal: 2
- of which minimal: 5
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.84488
- on average over minimal pinning sets: 2.52
- on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{1, 2, 5, 6} |
4 |
[2, 2, 2, 4] |
2.50 |
| B (optimal) |
• |
{1, 2, 5, 9} |
4 |
[2, 2, 2, 4] |
2.50 |
| a (minimal) |
• |
{1, 2, 5, 7, 8} |
5 |
[2, 2, 2, 3, 3] |
2.40 |
| b (minimal) |
• |
{1, 2, 3, 5, 8} |
5 |
[2, 2, 2, 3, 4] |
2.60 |
| c (minimal) |
• |
{1, 2, 4, 5, 7} |
5 |
[2, 2, 2, 3, 4] |
2.60 |
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 |
2.5 |
| 5 |
0 |
3 |
9 |
2.67 |
| 6 |
0 |
0 |
20 |
2.83 |
| 7 |
0 |
0 |
15 |
2.95 |
| 8 |
0 |
0 |
6 |
3.04 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
2 |
3 |
51 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 2, 2, 3, 3, 4, 4, 4, 4]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,5,6,6],[0,6,1,0],[1,6,5,5],[1,4,4,2],[2,4,3,2]]
- PD code (use to draw this multiloop with SnapPy): [[4,10,1,5],[5,9,6,8],[3,14,4,11],[9,1,10,2],[6,13,7,12],[7,11,8,12],[13,2,14,3]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (5,4,-6,-1)(14,3,-9,-4)(13,10,-14,-11)(2,9,-3,-10)(1,6,-2,-7)(12,7,-13,-8)(8,11,-5,-12)
- 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,-7,12,-5)(-2,-10,13,7)(-3,14,10)(-4,5,11,-14)(-6,1)(-8,-12)(-9,2,6,4)(-11,8,-13)(3,9)
Multiloop annotated with half-edges