$9^{1}_{1}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 7
- 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: 2.59028
- 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, 6, 8} |
7 |
[2, 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 |
| 7 |
1 |
0 |
0 |
2.0 |
| 8 |
0 |
0 |
2 |
2.62 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
1 |
0 |
3 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 2, 2, 2, 2, 2, 7, 7]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,1,2,2],[0,3,3,0],[0,4,4,0],[1,5,5,1],[2,6,6,2],[3,6,6,3],[4,5,5,4]]
- PD code (use to draw this loop with SnapPy): [[7,14,8,1],[13,6,14,7],[8,2,9,1],[5,12,6,13],[2,10,3,9],[11,4,12,5],[10,4,11,3]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (14,7,-1,-8)(8,1,-9,-2)(10,3,-11,-4)(6,13,-7,-14)(2,9,-3,-10)(4,11,-5,-12)(12,5,-13,-6)
- 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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,-10,-4,-12,-6,-14,-8)(-3,10)(-5,12)(-7,14)(-9,2)(-11,4)(-13,6)(1,7,13,5,11,3,9)
Loop annotated with half-edges