$12^{1}_{225}$ - Minimal pinning sets
Pinning sets for 12^1_225 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_225 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, 8, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 6, 8, 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, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,6],[0,7,7,8],[0,8,8,0],[1,9,9,5],[1,4,6,6],[1,5,5,7],[2,6,9,2],[2,9,3,3],[4,8,7,4]]
PD code (use to draw this loop with SnapPy): [[9,20,10,1],[5,8,6,9],[19,12,20,13],[10,2,11,1],[15,4,16,5],[16,7,17,8],[6,17,7,18],[13,18,14,19],[11,2,12,3],[3,14,4,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(13,4,-14,-5)(2,5,-3,-6)(16,9,-17,-10)(20,11,-1,-12)(3,14,-4,-15)(10,15,-11,-16)(6,17,-7,-18)(18,7,-19,-8)(8,19,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,-6,-18,-8,-20,-12)(-3,-15,10,-17,6)(-4,13,1,11,15)(-5,2,-13)(-7,18)(-9,16,-11,20)(-10,-16)(-14,3,5)(-19,8)(4,14)(7,17,9,19)
Loop annotated with half-edges
12^1_225 annotated with half-edges