$9^{2}_{7}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 4
- Total number of pinning sets: 72
- of which optimal: 2
- of which minimal: 6
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.99681
- on average over minimal pinning sets: 2.85
- on average over optimal pinning sets: 2.75
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{2, 4, 6, 8} |
4 |
[2, 3, 3, 3] |
2.75 |
| B (optimal) |
• |
{2, 3, 4, 8} |
4 |
[2, 3, 3, 3] |
2.75 |
| a (minimal) |
• |
{2, 4, 5, 7, 8} |
5 |
[2, 3, 3, 4, 4] |
3.20 |
| b (minimal) |
• |
{1, 3, 4, 8, 9} |
5 |
[2, 3, 3, 3, 3] |
2.80 |
| c (minimal) |
• |
{1, 3, 4, 6, 9} |
5 |
[2, 3, 3, 3, 3] |
2.80 |
| d (minimal) |
• |
{1, 2, 4, 6, 9} |
5 |
[2, 3, 3, 3, 3] |
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 |
2.75 |
| 5 |
0 |
4 |
9 |
2.89 |
| 6 |
0 |
0 |
26 |
2.99 |
| 7 |
0 |
0 |
22 |
3.05 |
| 8 |
0 |
0 |
8 |
3.09 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
2 |
4 |
66 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 4, 4]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,6],[0,6,5,1],[1,4,6,2],[3,5,4,3]]
- PD code (use to draw this multiloop with SnapPy): [[4,14,1,5],[5,11,6,10],[3,9,4,10],[13,8,14,9],[1,12,2,11],[6,2,7,3],[7,12,8,13]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (5,4,-6,-1)(12,7,-13,-8)(1,8,-2,-9)(9,14,-10,-5)(10,3,-11,-4)(6,11,-7,-12)(2,13,-3,-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,-9,-5)(-2,-14,9)(-3,10,14)(-4,5,-10)(-6,-12,-8,1)(-7,12)(-11,6,4)(-13,2,8)(3,13,7,11)
Multiloop annotated with half-edges