$9^{1}_{4}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this loop: 3
- Total number of pinning sets: 64
- 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: 2.79396
- 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} |
3 |
[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 |
| 3 |
1 |
0 |
0 |
2.0 |
| 4 |
0 |
0 |
6 |
2.42 |
| 5 |
0 |
0 |
15 |
2.67 |
| 6 |
0 |
0 |
20 |
2.83 |
| 7 |
0 |
0 |
15 |
2.95 |
| 8 |
0 |
0 |
6 |
3.04 |
| 9 |
0 |
0 |
1 |
3.11 |
| Total |
1 |
0 |
63 |
|
Other information about this loop
Properties
- Region degree sequence: [2, 2, 2, 3, 3, 4, 4, 4, 4]
- Minimal region degree: 2
- Is multisimple: No
Combinatorial encoding data
- Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,5,4,3],[0,2,1,0],[1,2,6,6],[1,6,6,2],[4,5,5,4]]
- PD code (use to draw this loop with SnapPy): [[7,14,8,1],[9,6,10,7],[13,2,14,3],[8,2,9,1],[5,12,6,13],[10,4,11,3],[11,4,12,5]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (14,7,-1,-8)(8,1,-9,-2)(6,3,-7,-4)(2,9,-3,-10)(4,11,-5,-12)(12,5,-13,-6)(10,13,-11,-14)
- 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,-14,-8)(-3,6,-13,10)(-4,-12,-6)(-5,12)(-7,14,-11,4)(-9,2)(1,7,3,9)(5,11,13)
Loop annotated with half-edges