$11^{1}_{97}$ - Minimal pinning sets
Pinning sets for 11^1_97 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_97 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 395
of which optimal: 2
of which minimal: 13
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.23145
on average over minimal pinning sets: 3.24615
on average over optimal pinning sets: 3.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 7, 8}
4
[3, 3, 3, 3]
3.00
B (optimal)
• {1, 3, 6, 10}
4
[3, 3, 3, 3]
3.00
a (minimal)
• {1, 2, 6, 7, 11}
5
[3, 3, 3, 3, 4]
3.20
b (minimal)
• {2, 3, 4, 6, 9}
5
[3, 3, 3, 3, 4]
3.20
c (minimal)
• {2, 5, 6, 8, 10}
5
[3, 3, 3, 3, 4]
3.20
d (minimal)
• {2, 5, 6, 9, 11}
5
[3, 3, 4, 4, 4]
3.60
e (minimal)
• {2, 4, 6, 7, 9, 11}
6
[3, 3, 3, 3, 4, 4]
3.33
f (minimal)
• {1, 3, 4, 7, 8, 10}
6
[3, 3, 3, 3, 3, 3]
3.00
g (minimal)
• {1, 2, 3, 6, 9, 11}
6
[3, 3, 3, 3, 4, 4]
3.33
h (minimal)
• {2, 5, 6, 7, 8, 11}
6
[3, 3, 3, 3, 4, 4]
3.33
i (minimal)
• {2, 4, 5, 6, 8, 9}
6
[3, 3, 3, 3, 4, 4]
3.33
j (minimal)
• {1, 2, 5, 6, 10, 11}
6
[3, 3, 3, 3, 4, 4]
3.33
k (minimal)
• {2, 3, 5, 6, 9, 10}
6
[3, 3, 3, 3, 4, 4]
3.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
3.0
5
0
4
14
3.13
6
0
7
66
3.21
7
0
0
130
3.23
8
0
0
111
3.25
9
0
0
49
3.27
10
0
0
11
3.27
11
0
0
1
3.27
Total
2
11
382
Other information about this loop Properties
Region degree sequence: [3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4]
Minimal region degree: 3
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,7],[0,7,8,1],[1,8,6,2],[3,5,8,7],[3,6,8,4],[4,7,6,5]]
PD code (use to draw this loop with SnapPy): [[18,5,1,6],[6,11,7,12],[12,17,13,18],[13,4,14,5],[1,10,2,11],[7,16,8,17],[8,3,9,4],[14,9,15,10],[2,15,3,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(11,2,-12,-3)(16,3,-17,-4)(12,7,-13,-8)(17,8,-18,-9)(4,9,-5,-10)(18,13,-1,-14)(5,14,-6,-15)(10,15,-11,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,14)(-2,11,15,-6)(-3,16,-11)(-4,-10,-16)(-5,-15,10)(-7,12,2)(-8,17,3,-12)(-9,4,-17)(-13,18,8)(-14,5,9,-18)(1,13,7)
Loop annotated with half-edges
11^1_97 annotated with half-edges