$11^{1}_{71}$ - Minimal pinning sets
Pinning sets for 11^1_71 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_71 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 136
of which optimal: 1
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04344
on average over minimal pinning sets: 2.65714
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, 7, 8}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 4, 5, 7, 8}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 2, 4, 6, 8, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 3, 4, 6, 8, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
d (minimal)
• {1, 3, 5, 6, 8, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
e (minimal)
• {1, 2, 4, 6, 7, 8}
6
[2, 2, 3, 3, 3, 3]
2.67
f (minimal)
• {1, 3, 4, 6, 7, 8}
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
6
6
2.74
7
0
0
36
2.93
8
0
0
47
3.08
9
0
0
30
3.19
10
0
0
9
3.24
11
0
0
1
3.27
Total
1
6
129
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 3, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,2],[0,1,5,3],[0,2,6,0],[1,7,8,5],[1,4,8,2],[3,8,7,7],[4,6,6,8],[4,7,6,5]]
PD code (use to draw this loop with SnapPy): [[11,18,12,1],[10,3,11,4],[17,2,18,3],[12,2,13,1],[4,15,5,16],[16,9,17,10],[13,6,14,7],[7,14,8,15],[5,8,6,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,18,-16,-1)(8,1,-9,-2)(11,4,-12,-5)(3,6,-4,-7)(14,7,-15,-8)(5,12,-6,-13)(2,13,-3,-14)(9,16,-10,-17)(17,10,-18,-11)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-15)(-2,-14,-8)(-3,-7,14)(-4,11,-18,15,7)(-5,-13,2,-9,-17,-11)(-6,3,13)(-10,17)(-12,5)(-16,9,1)(4,6,12)(10,16,18)
Loop annotated with half-edges
11^1_71 annotated with half-edges