$11^{1}_{50}$ - Minimal pinning sets
Pinning sets for 11^1_50 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_50 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 224
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05039
on average over minimal pinning sets: 2.62
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, 3, 4, 10}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 3, 5, 9, 10}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 3, 5, 8, 10}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 2, 3, 5, 8}
5
[2, 2, 3, 3, 3]
2.60
d (minimal)
• {1, 2, 3, 4, 8}
5
[2, 2, 3, 3, 3]
2.60
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
4
7
2.71
6
0
0
39
2.89
7
0
0
67
3.03
8
0
0
63
3.13
9
0
0
33
3.2
10
0
0
9
3.24
11
0
0
1
3.27
Total
1
4
219
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: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,7,8],[0,8,6,5],[1,4,2,1],[2,4,8,7],[3,6,8,3],[3,7,6,4]]
PD code (use to draw this loop with SnapPy): [[18,5,1,6],[6,16,7,15],[17,14,18,15],[11,4,12,5],[1,9,2,8],[16,8,17,7],[2,13,3,14],[3,10,4,11],[12,10,13,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(15,2,-16,-3)(4,13,-5,-14)(5,16,-6,-17)(6,9,-7,-10)(18,7,-1,-8)(17,10,-18,-11)(14,11,-15,-12)(12,3,-13,-4)
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)(-2,15,11,-18,-8)(-3,12,-15)(-4,-14,-12)(-5,-17,-11,14)(-6,-10,17)(-7,18,10)(-9,6,16,2)(-13,4)(-16,5,13,3)(1,7,9)
Loop annotated with half-edges
11^1_50 annotated with half-edges