$11^{2}_{5}$ - Minimal pinning sets
Pinning sets for 11^2_5 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_5 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 250
of which optimal: 5
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97956
on average over minimal pinning sets: 2.48333
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)
• {3, 5, 8, 9}
4
[2, 2, 2, 5]
2.75
B (optimal)
• {3, 5, 6, 8}
4
[2, 2, 2, 3]
2.25
C (optimal)
• {3, 5, 8, 10}
4
[2, 2, 2, 4]
2.50
D (optimal)
• {3, 5, 7, 8}
4
[2, 2, 2, 4]
2.50
E (optimal)
• {2, 3, 5, 8}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {1, 3, 4, 5, 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
4
5
0
0
2.5
5
0
1
25
2.71
6
0
0
56
2.88
7
0
0
70
3.0
8
0
0
56
3.09
9
0
0
28
3.17
10
0
0
8
3.23
11
0
0
1
3.27
Total
5
1
244
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,6,7],[0,7,8,5],[0,5,1,1],[1,4,3,6],[2,5,8,2],[2,8,8,3],[3,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[3,14,4,1],[2,9,3,10],[13,18,14,15],[4,7,5,8],[1,11,2,10],[11,8,12,9],[15,12,16,13],[6,17,7,18],[5,17,6,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (2,5,-3,-6)(9,6,-10,-7)(14,7,-1,-8)(8,13,-9,-14)(17,10,-18,-11)(1,12,-2,-13)(18,3,-15,-4)(4,15,-5,-16)(11,16,-12,-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,-13,8)(-2,-6,9,13)(-3,18,10,6)(-4,-16,11,-18)(-5,2,12,16)(-7,14,-9)(-8,-14)(-10,17,-12,1,7)(-11,-17)(-15,4)(3,5,15)
Multiloop annotated with half-edges
11^2_5 annotated with half-edges