$11^{1}_{78}$ - Minimal pinning sets
Pinning sets for 11^1_78 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_78 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 326
of which optimal: 1
of which minimal: 13
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.15002
on average over minimal pinning sets: 2.94872
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, 6, 10}
4
[2, 3, 3, 4]
3.00
a (minimal)
• {1, 3, 6, 8, 11}
5
[2, 3, 3, 3, 4]
3.00
b (minimal)
• {2, 3, 6, 7, 11}
5
[2, 3, 3, 3, 4]
3.00
c (minimal)
• {1, 3, 6, 9, 10}
5
[2, 3, 3, 3, 4]
3.00
d (minimal)
• {1, 3, 5, 6, 11}
5
[2, 3, 3, 3, 3]
2.80
e (minimal)
• {2, 3, 5, 6, 11}
5
[2, 3, 3, 3, 3]
2.80
f (minimal)
• {1, 2, 5, 6, 10}
5
[2, 3, 3, 3, 3]
2.80
g (minimal)
• {1, 3, 5, 6, 10}
5
[2, 3, 3, 3, 3]
2.80
h (minimal)
• {2, 5, 6, 10, 11}
5
[2, 3, 3, 3, 3]
2.80
i (minimal)
• {1, 2, 3, 6, 10, 11}
6
[2, 3, 3, 3, 3, 3]
2.83
j (minimal)
• {2, 3, 4, 6, 8, 11}
6
[2, 3, 3, 3, 4, 4]
3.17
k (minimal)
• {1, 3, 4, 6, 8, 10}
6
[2, 3, 3, 3, 4, 4]
3.17
l (minimal)
• {1, 3, 6, 7, 9, 11}
6
[2, 3, 3, 3, 4, 4]
3.17
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
3.0
5
0
8
7
2.97
6
0
4
59
3.06
7
0
0
106
3.14
8
0
0
89
3.2
9
0
0
41
3.24
10
0
0
10
3.26
11
0
0
1
3.27
Total
1
12
313
Other information about this loop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,5,6],[0,6,7,0],[0,7,8,4],[1,3,8,5],[1,4,8,6],[1,5,7,2],[2,6,8,3],[3,7,5,4]]
PD code (use to draw this loop with SnapPy): [[5,18,6,1],[4,13,5,14],[17,6,18,7],[1,11,2,10],[14,10,15,9],[3,8,4,9],[12,7,13,8],[16,11,17,12],[2,16,3,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,4,-16,-5)(10,5,-11,-6)(1,6,-2,-7)(7,12,-8,-13)(8,17,-9,-18)(14,9,-15,-10)(2,11,-3,-12)(13,18,-14,-1)(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,-13)(-2,-12,7)(-3,-17,8,12)(-4,15,9,17)(-5,10,-15)(-6,1,-14,-10)(-8,-18,13)(-9,14,18)(-11,2,6)(-16,3,11,5)(4,16)
Loop annotated with half-edges
11^1_78 annotated with half-edges