$8^{3}_{2}$ - Minimal pinning sets
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning data
- Pinning number of this multiloop: 4
- Total number of pinning sets: 35
- of which optimal: 2
- of which minimal: 6
- The mean region-degree (mean-degree) of a pinning set is
- on average over all pinning sets: 3.0
- on average over minimal pinning sets: 3.0
- on average over optimal pinning sets: 3.0
Refined data for the minimal pinning sets
Pin label |
Pin color |
Regions |
Cardinality |
Degree sequence |
Mean-degree |
A (optimal) |
• |
{1, 4, 5, 8} |
4 |
[3, 3, 3, 3] |
3.00 |
B (optimal) |
• |
{2, 3, 6, 7} |
4 |
[3, 3, 3, 3] |
3.00 |
a (minimal) |
• |
{1, 2, 3, 5, 6, 8} |
6 |
[3, 3, 3, 3, 3, 3] |
3.00 |
b (minimal) |
• |
{1, 2, 4, 5, 6, 7} |
6 |
[3, 3, 3, 3, 3, 3] |
3.00 |
c (minimal) |
• |
{2, 3, 4, 5, 7, 8} |
6 |
[3, 3, 3, 3, 3, 3] |
3.00 |
d (minimal) |
• |
{1, 3, 4, 6, 7, 8} |
6 |
[3, 3, 3, 3, 3, 3] |
3.00 |
Data for pinning sets in each cardinal
Cardinality |
Optimal pinning sets |
Minimal suboptimal pinning sets |
Nonminimal pinning sets |
Averaged mean-degree |
4 |
2 |
0 |
0 |
3.0 |
5 |
0 |
0 |
8 |
3.0 |
6 |
0 |
4 |
12 |
3.0 |
7 |
0 |
0 |
8 |
3.0 |
8 |
0 |
0 |
1 |
3.0 |
Total |
2 |
4 |
29 |
|
Other information about this multiloop
Properties
- Region degree sequence: [3, 3, 3, 3, 3, 3, 3, 3]
- Minimal region degree: 3
- Is multisimple: Yes
Combinatorial encoding data
- Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,5,4],[0,3,5,1],[1,4,3,2]]
- PD code (use to draw this multiloop with SnapPy): [[4,8,1,5],[5,9,6,12],[3,11,4,12],[7,10,8,11],[1,10,2,9],[6,2,7,3]]
- Permutation representation (action on half-edges):
- Vertex permutation $\sigma=$ (12,3,-9,-4)(1,10,-2,-11)(9,8,-10,-5)(4,5,-1,-6)(6,11,-7,-12)(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,-11,6)(-2,7,11)(-3,12,-7)(-4,-6,-12)(-5,4,-9)(-8,9,3)(-10,1,5)(2,10,8)
Multiloop annotated with half-edges