$11^{1}_{58}$ - Minimal pinning sets
Pinning sets for 11^1_58 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_58 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 168
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98725
on average over minimal pinning sets: 2.5
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, 6, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 6, 8, 10}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 3, 6, 7, 10}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 2, 3, 6, 8}
5
[2, 2, 2, 3, 3]
2.40
E (optimal)
• {1, 2, 3, 6, 7}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 3, 4, 6, 8}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
6
0
0
2.5
6
0
0
27
2.77
7
0
0
50
2.95
8
0
0
49
3.07
9
0
0
27
3.16
10
0
0
8
3.23
11
0
0
1
3.27
Total
6
0
162
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 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],[0,2,6,7],[1,7,7,8],[2,8,8,6],[3,5,8,7],[3,6,4,4],[4,6,5,5]]
PD code (use to draw this loop with SnapPy): [[18,11,1,12],[12,17,13,18],[13,10,14,11],[1,14,2,15],[7,16,8,17],[4,9,5,10],[2,5,3,6],[15,6,16,7],[8,3,9,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(18,3,-1,-4)(13,4,-14,-5)(2,7,-3,-8)(11,8,-12,-9)(16,9,-17,-10)(17,12,-18,-13)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,14,4)(-2,-8,11,15,-6)(-3,18,12,8)(-4,13,-18)(-5,-15,10,-17,-13)(-7,2)(-9,16,-11)(-10,-16)(-12,17,9)(-14,5)(1,3,7)
Loop annotated with half-edges
11^1_58 annotated with half-edges