$8^{2}_{1}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 6
- Total number of pinning sets: 4
- 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: 2.53571
- 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, 3, 4, 5, 7} |
6 |
[2, 2, 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 |
| 6 |
1 |
0 |
0 |
2.0 |
| 7 |
0 |
0 |
2 |
2.57 |
| 8 |
0 |
0 |
1 |
3.0 |
| Total |
1 |
0 |
3 |
|
Other information about this multiloop
Properties
- Region degree sequence: [2, 2, 2, 2, 2, 2, 6, 6]
- Minimal region degree: 2
- Is multisimple: Yes
Combinatorial encoding data
- Plantri embedding: [[1,1,2,2],[0,3,3,0],[0,4,4,0],[1,5,5,1],[2,5,5,2],[3,4,4,3]]
- PD code (use to draw this multiloop with SnapPy): [[6,12,1,7],[7,5,8,6],[11,1,12,2],[4,8,5,9],[2,10,3,11],[9,3,10,4]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (12,1,-7,-2)(10,3,-11,-4)(8,5,-9,-6)(6,7,-1,-8)(4,9,-5,-10)(2,11,-3,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3,10,-5,8)(-2,-12)(-4,-10)(-6,-8)(-7,6,-9,4,-11,2)(1,7)(3,11)(5,9)
Multiloop annotated with half-edges