$11^{1}_{56}$ - Minimal pinning sets
Pinning sets for 11^1_56 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_56 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 256
of which optimal: 1
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0605
on average over minimal pinning sets: 2.7125
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 6, 10}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 5, 8}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 2, 4, 6, 8}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
d (minimal)
• {1, 4, 5, 8, 10}
5
[2, 2, 3, 3, 4]
2.80
e (minimal)
• {1, 4, 5, 7, 10}
5
[2, 2, 3, 3, 4]
2.80
f (minimal)
• {1, 4, 5, 9, 10}
5
[2, 2, 3, 3, 3]
2.60
g (minimal)
• {1, 2, 4, 6, 11}
5
[2, 2, 3, 3, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.5
5
0
7
7
2.74
6
0
0
49
2.93
7
0
0
79
3.05
8
0
0
69
3.14
9
0
0
34
3.2
10
0
0
9
3.24
11
0
0
1
3.27
Total
1
7
248
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,3],[0,2,6,4],[1,3,6,7],[2,7,8,6],[3,5,8,4],[4,8,8,5],[5,7,7,6]]
PD code (use to draw this loop with SnapPy): [[9,18,10,1],[17,8,18,9],[10,8,11,7],[1,7,2,6],[16,5,17,6],[11,14,12,15],[2,15,3,16],[13,4,14,5],[12,4,13,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,9,-1,-10)(10,1,-11,-2)(13,2,-14,-3)(3,16,-4,-17)(4,7,-5,-8)(14,5,-15,-6)(8,11,-9,-12)(17,12,-18,-13)(6,15,-7,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,13,-18,-10)(-3,-17,-13)(-4,-8,-12,17)(-5,14,2,-11,8)(-6,-16,3,-14)(-7,4,16)(-9,18,12)(-15,6)(1,9,11)(5,7,15)
Loop annotated with half-edges
11^1_56 annotated with half-edges