$12^{1}_{69}$ - Minimal pinning sets
Pinning sets for 12^1_69 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_69 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 160
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97043
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 6, 8, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
1
7
2.5
7
0
0
26
2.74
8
0
0
45
2.92
9
0
0
45
3.07
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
158
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,2],[0,3,4,0],[0,4,3,0],[1,2,5,6],[1,7,7,2],[3,7,8,6],[3,5,8,9],[4,9,5,4],[5,9,9,6],[6,8,8,7]]
PD code (use to draw this loop with SnapPy): [[20,11,1,12],[12,19,13,20],[10,1,11,2],[18,9,19,10],[13,3,14,2],[4,17,5,18],[5,8,6,9],[3,15,4,14],[7,16,8,17],[6,16,7,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,11,-1,-12)(3,6,-4,-7)(16,7,-17,-8)(18,9,-19,-10)(12,1,-13,-2)(13,4,-14,-5)(5,14,-6,-15)(2,15,-3,-16)(10,17,-11,-18)(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,-16,-8,-20,-12)(-3,-7,16)(-4,13,1,11,17,7)(-5,-15,2,-13)(-6,3,15)(-9,18,-11,20)(-10,-18)(-14,5)(-17,10,-19,8)(4,6,14)(9,19)
Loop annotated with half-edges
12^1_69 annotated with half-edges