$12^{1}_{269}$ - Minimal pinning sets
Pinning sets for 12^1_269
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_269
Pinning data
Pinning number of this loop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 3, 4, 6, 7, 11}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
•
{1, 3, 4, 6, 7, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
Other information about this loop
Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,4,0],[0,4,5,0],[1,6,6,7],[1,8,5,2],[2,4,8,9],[3,9,9,3],[3,9,8,8],[4,7,7,5],[5,7,6,6]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[19,10,20,11],[12,2,13,1],[18,5,19,6],[9,2,10,3],[13,9,14,8],[6,17,7,18],[4,15,5,16],[3,15,4,14],[16,7,17,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,2,-16,-3)(13,4,-14,-5)(5,10,-6,-11)(16,7,-17,-8)(11,20,-12,-1)(1,12,-2,-13)(3,14,-4,-15)(8,17,-9,-18)(18,9,-19,-10)(6,19,-7,-20)
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)(-15,15)(-16,16)(-17,17)(-18,18)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-13,-5,-11)(-2,15,-4,13)(-3,-15)(-6,-20,11)(-7,16,2,12,20)(-8,-18,-10,5,-14,3,-16)(-9,18)(-12,1)(-17,8)(-19,6,10)(4,14)(7,19,9,17)
Loop annotated with half-edges
12^1_269 annotated with half-edges