$11^{2}_{38}$ - Minimal pinning sets
Pinning sets for 11^2_38 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_38 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 166
of which optimal: 5
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98544
on average over minimal pinning sets: 2.57619
on average over optimal pinning sets: 2.44
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 4, 8, 10}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 2, 4, 5, 10}
5
[2, 2, 2, 3, 3]
2.40
E (optimal)
• {1, 2, 3, 4, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 4, 5, 7, 8, 10}
6
[2, 2, 2, 3, 4, 4]
2.83
b (minimal)
• {1, 4, 5, 8, 10, 11}
6
[2, 2, 2, 3, 4, 5]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
5
0
0
2.44
6
0
2
24
2.74
7
0
0
50
2.94
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
5
2
159
Other information about this multiloop 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: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,7,4],[0,3,8,5],[1,4,2,1],[2,8,8,7],[3,6,8,3],[4,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[8,18,1,9],[9,6,10,5],[7,4,8,5],[17,12,18,13],[1,12,2,11],[6,11,7,10],[15,3,16,4],[13,16,14,17],[2,14,3,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(5,2,-6,-3)(3,14,-4,-15)(15,4,-16,-5)(16,7,-17,-8)(12,17,-13,-18)(8,9,-1,-10)(18,11,-9,-12)(6,13,-7,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,5,-16,-8,-10)(-3,-15,-5)(-4,15)(-6,-14,3)(-7,16,4,14)(-9,8,-17,12)(-11,18,-13,6,2)(-12,-18)(1,9,11)(7,13,17)
Multiloop annotated with half-edges
11^2_38 annotated with half-edges