$12^{1}_{208}$ - Minimal pinning sets
Pinning sets for 12^1_208 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^1_208 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 308
of which optimal: 1
of which minimal: 11
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10286
on average over minimal pinning sets: 2.70606
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 5, 8, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 3, 4, 5, 8, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 3, 4, 5, 6, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 2, 3, 5, 8, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
d (minimal)
• {1, 2, 3, 5, 6, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
e (minimal)
• {1, 3, 4, 5, 8, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
f (minimal)
• {1, 3, 4, 5, 6, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
g (minimal)
• {2, 3, 5, 6, 10, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
h (minimal)
• {2, 3, 5, 6, 9, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
i (minimal)
• {2, 3, 5, 6, 7, 11}
6
[2, 2, 3, 3, 3, 5]
3.00
j (minimal)
• {2, 3, 4, 5, 6, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
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
2.6
6
0
10
7
2.75
7
0
0
60
2.95
8
0
0
96
3.09
9
0
0
83
3.2
10
0
0
40
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
10
297
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,8,4],[0,3,8,5],[1,4,2,1],[2,9,9,7],[3,6,9,8],[3,7,9,4],[6,8,7,6]]
PD code (use to draw this loop with SnapPy): [[11,20,12,1],[10,17,11,18],[19,16,20,17],[12,7,13,8],[1,8,2,9],[18,9,19,10],[15,4,16,5],[6,3,7,4],[13,3,14,2],[5,14,6,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,11,-1,-12)(4,1,-5,-2)(13,2,-14,-3)(10,5,-11,-6)(19,6,-20,-7)(16,7,-17,-8)(3,12,-4,-13)(17,14,-18,-15)(8,15,-9,-16)(9,18,-10,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,12)(-2,13,-4)(-3,-13)(-5,10,18,14,2)(-6,19,-10)(-7,16,-9,-19)(-8,-16)(-11,20,6)(-12,3,-14,17,7,-20)(-15,8,-17)(-18,9,15)(1,11,5)
Loop annotated with half-edges
12^1_208 annotated with half-edges