$11^{2}_{43}$ - Minimal pinning sets
Pinning sets for 11^2_43 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_43 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 180
of which optimal: 6
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.99776
on average over minimal pinning sets: 2.57143
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, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 5, 8, 9}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 2, 3, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 3, 7, 8}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {1, 2, 3, 6, 8}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 3, 6, 8, 10}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 3, 5, 7, 8, 10}
6
[2, 2, 2, 4, 4, 4]
3.00
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
1
28
2.78
7
0
0
55
2.96
8
0
0
53
3.08
9
0
0
28
3.17
10
0
0
8
3.23
11
0
0
1
3.27
Total
6
1
173
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],[0,5,5,6],[0,6,3,3],[0,2,2,7],[0,7,8,5],[1,4,6,1],[1,5,8,2],[3,8,8,4],[4,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[7,10,8,1],[6,18,7,11],[13,9,14,10],[8,14,9,15],[1,4,2,5],[11,5,12,6],[12,17,13,18],[15,3,16,4],[2,16,3,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-11,-2)(9,2,-10,-3)(14,3,-15,-4)(16,7,-17,-8)(5,8,-6,-9)(6,17,-7,-18)(10,11,-1,-12)(15,12,-16,-13)(4,13,-5,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,-7,16,12)(-2,9,-6,-18)(-3,14,-5,-9)(-4,-14)(-8,5,13,-16)(-10,-12,15,3)(-11,10,2)(-13,4,-15)(-17,6,8)(1,11)(7,17)
Multiloop annotated with half-edges
11^2_43 annotated with half-edges