$11^{1}_{54}$ - Minimal pinning sets
Pinning sets for 11^1_54 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_54 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 168
of which optimal: 2
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05812
on average over minimal pinning sets: 2.76296
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)
• {1, 3, 5, 6, 10}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {2, 3, 7, 10, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 3, 5, 7, 8, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
b (minimal)
• {1, 3, 4, 6, 7, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
c (minimal)
• {1, 3, 4, 7, 8, 10}
6
[2, 2, 3, 3, 4, 4]
3.00
d (minimal)
• {1, 2, 3, 6, 7, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
e (minimal)
• {1, 2, 3, 7, 8, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
f (minimal)
• {2, 3, 5, 6, 10, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
g (minimal)
• {1, 3, 4, 7, 10, 11}
6
[2, 2, 3, 3, 3, 4]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.6
6
0
7
12
2.81
7
0
0
49
2.99
8
0
0
56
3.11
9
0
0
32
3.19
10
0
0
9
3.24
11
0
0
1
3.27
Total
2
7
159
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,7,3],[0,2,7,7],[0,8,8,5],[1,4,8,6],[1,5,8,7],[2,6,3,3],[4,6,5,4]]
PD code (use to draw this loop with SnapPy): [[5,18,6,1],[13,4,14,5],[14,17,15,18],[6,15,7,16],[1,11,2,10],[12,9,13,10],[3,8,4,9],[16,7,17,8],[11,3,12,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,9,-1,-10)(11,2,-12,-3)(14,5,-15,-6)(1,6,-2,-7)(10,7,-11,-8)(8,17,-9,-18)(4,13,-5,-14)(12,15,-13,-16)(3,16,-4,-17)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-7,10)(-2,11,7)(-3,-17,8,-11)(-4,-14,-6,1,9,17)(-5,14)(-8,-18,-10)(-9,18)(-12,-16,3)(-13,4,16)(-15,12,2,6)(5,13,15)
Loop annotated with half-edges
11^1_54 annotated with half-edges