$11^{2}_{25}$ - Minimal pinning sets
Pinning sets for 11^2_25 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_25 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 158
of which optimal: 7
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97783
on average over minimal pinning sets: 2.54286
on average over optimal pinning sets: 2.54286
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 6, 8}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 5, 6, 8}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 3, 6, 8, 10}
5
[2, 2, 2, 3, 6]
3.00
E (optimal)
• {1, 3, 6, 7, 8}
5
[2, 2, 2, 3, 5]
2.80
F (optimal)
• {1, 2, 4, 6, 8}
5
[2, 2, 2, 3, 3]
2.40
G (optimal)
• {1, 3, 4, 6, 8}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
7
0
0
2.54
6
0
0
26
2.77
7
0
0
45
2.93
8
0
0
45
3.06
9
0
0
26
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
7
0
151
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,2,2],[0,1,1,3],[0,2,5,6],[0,7,8,1],[3,8,6,6],[3,5,5,7],[4,6,8,8],[4,7,7,5]]
PD code (use to draw this multiloop with SnapPy): [[14,11,1,12],[12,5,13,6],[6,13,7,14],[7,10,8,11],[1,4,2,5],[9,18,10,15],[8,18,9,17],[3,16,4,17],[2,16,3,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(9,2,-10,-3)(4,7,-5,-8)(11,8,-12,-9)(3,10,-4,-11)(18,13,-15,-14)(14,15,-1,-16)(16,5,-17,-6)(6,17,-7,-18)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,8,-5,16)(-2,9,-12)(-3,-11,-9)(-4,-8,11)(-6,-18,-14,-16)(-7,4,10,2,-13,18)(-10,3)(-15,14)(-17,6)(1,15,13)(5,7,17)
Multiloop annotated with half-edges
11^2_25 annotated with half-edges