$8^{1}_{3}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 4
- Total number of pinning sets: 32
- of which optimal: 3
- of which minimal: 3
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 2.80461
- on average over minimal pinning sets: 2.5
- on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
| Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
| A (optimal) |
• |
{1, 2, 3, 7} |
4 |
[2, 2, 3, 3] |
2.50 |
| B (optimal) |
• |
{1, 4, 5, 7} |
4 |
[2, 2, 3, 3] |
2.50 |
| C (optimal) |
• |
{1, 3, 4, 7} |
4 |
[2, 2, 3, 3] |
2.50 |
Data for pinning sets in each cardinal
| Cardinality |
Optimal pinning sets |
Minimal suboptimal pinning sets |
Nonminimal pinning sets |
Averaged mean-degree |
| 4 |
3 |
0 |
0 |
2.5 |
| 5 |
0 |
0 |
10 |
2.72 |
| 6 |
0 |
0 |
12 |
2.86 |
| 7 |
0 |
0 |
6 |
2.95 |
| 8 |
0 |
0 |
1 |
3.0 |
| Total |
3 |
0 |
29 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,3],[0,2,4,4],[0,3,3,5],[1,4,2,1]]
- PD code (use to draw this loop with SnapPy): [[12,3,1,4],[4,10,5,9],[11,8,12,9],[2,7,3,8],[1,7,2,6],[10,6,11,5]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (10,1,-11,-2)(3,8,-4,-9)(4,11,-5,-12)(12,5,-1,-6)(9,6,-10,-7)(7,2,-8,-3)
- 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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,6)(-2,7,-10)(-3,-9,-7)(-4,-12,-6,9)(-5,12)(-8,3)(-11,4,8,2)(1,5,11)
Loop annotated with half-edges