$8^{3}_{1}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 4
- Total number of pinning sets: 25
- 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: 2.72286
- on average over minimal pinning sets: 2.5
- 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, 5, 7} |
4 |
[2, 2, 2, 3] |
2.25 |
| B (optimal) |
• |
{1, 2, 3, 7} |
4 |
[2, 2, 2, 3] |
2.25 |
| a (minimal) |
• |
{1, 3, 4, 6, 7, 8} |
6 |
[2, 2, 2, 4, 4, 4] |
3.00 |
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 |
0 |
7 |
2.57 |
| 6 |
0 |
1 |
9 |
2.8 |
| 7 |
0 |
0 |
5 |
2.91 |
| 8 |
0 |
0 |
1 |
3.0 |
| Total |
2 |
1 |
22 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 2, 2, 3, 3, 4, 4, 4]
- Minimal region degree: 2
- Is multisimple: Yes
Combinatorial encoding data
- Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,5],[0,5,4,0],[1,3,5,1],[2,4,3,2]]
- PD code (use to draw this multiloop with SnapPy): [[4,8,1,5],[5,9,6,12],[3,11,4,12],[7,1,8,2],[9,7,10,6],[10,2,11,3]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (9,2,-10,-3)(8,3,-5,-4)(12,7,-9,-8)(1,10,-2,-11)(4,5,-1,-6)(6,11,-7,-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)
- Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-11,6)(-2,9,7,11)(-3,8,-9)(-4,-6,-12,-8)(-5,4)(-7,12)(-10,1,5,3)(2,10)
Multiloop annotated with half-edges