$8^{2}_{2}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 4
- Total number of pinning sets: 16
- of which optimal: 1
- of which minimal: 1
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.62679
- on average over minimal pinning sets: 2.0
- on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{1, 2, 4, 5} |
4 |
[2, 2, 2, 2] |
2.00 |
Data for pinning sets in each cardinal
| Cardinality |
Optimal pinning sets |
Minimal suboptimal pinning sets |
Nonminimal pinning sets |
Averaged mean-degree |
| 4 |
1 |
0 |
0 |
2.0 |
| 5 |
0 |
0 |
4 |
2.4 |
| 6 |
0 |
0 |
6 |
2.67 |
| 7 |
0 |
0 |
4 |
2.86 |
| 8 |
0 |
0 |
1 |
3.0 |
| Total |
1 |
0 |
15 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 2, 2, 2, 4, 4, 4, 4]
- Minimal region degree: 2
- Is multisimple: Yes
Combinatorial encoding data
- Plantri embedding: [[1,1,2,3],[0,4,4,0],[0,4,5,5],[0,5,5,4],[1,3,2,1],[2,3,3,2]]
- PD code (use to draw this multiloop with SnapPy): [[6,12,1,7],[7,5,8,6],[11,3,12,4],[1,10,2,9],[4,8,5,9],[2,10,3,11]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (8,1,-9,-2)(10,3,-11,-4)(2,11,-3,-12)(12,5,-7,-6)(6,7,-1,-8)(4,9,-5,-10)
- 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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,-12,-6,-8)(-3,10,-5,12)(-4,-10)(-7,6)(-9,4,-11,2)(1,7,5,9)(3,11)
Multiloop annotated with half-edges