$9^{2}_{4}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 4
- Total number of pinning sets: 58
- of which optimal: 3
- of which minimal: 4
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.8304
- on average over minimal pinning sets: 2.4625
- on average over optimal pinning sets: 2.41667
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{1, 3, 5, 8} |
4 |
[2, 2, 2, 3] |
2.25 |
| B (optimal) |
• |
{1, 5, 8, 9} |
4 |
[2, 2, 2, 4] |
2.50 |
| C (optimal) |
• |
{1, 4, 5, 8} |
4 |
[2, 2, 2, 4] |
2.50 |
| a (minimal) |
• |
{1, 2, 5, 6, 8} |
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 |
3 |
0 |
0 |
2.42 |
| 5 |
0 |
1 |
12 |
2.66 |
| 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 |
3 |
1 |
54 |
|
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: [[1,2,2,3],[0,4,5,2],[0,1,6,0],[0,6,4,4],[1,3,3,5],[1,4,6,6],[2,5,5,3]]
- PD code (use to draw this multiloop with SnapPy): [[10,5,1,6],[6,3,7,4],[4,9,5,10],[1,11,2,14],[2,13,3,14],[7,13,8,12],[8,11,9,12]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (4,1,-5,-2)(8,3,-9,-4)(2,7,-3,-8)(14,5,-11,-6)(10,11,-1,-12)(12,9,-13,-10)(6,13,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,-9,12)(-2,-8,-4)(-3,8)(-5,14,-7,2)(-6,-14)(-10,-12)(-11,10,-13,6)(1,11,5)(3,7,13,9)
Multiloop annotated with half-edges