$11^{3}_{19}$ - Minimal pinning sets
Pinning sets for 11^3_19 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^3_19 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 307
of which optimal: 2
of which minimal: 15
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.09372
on average over minimal pinning sets: 2.93111
on average over optimal pinning sets: 2.75
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 9}
4
[2, 2, 3, 4]
2.75
B (optimal)
• {1, 4, 6, 10}
4
[2, 2, 3, 4]
2.75
a (minimal)
• {1, 3, 4, 7, 8}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 4, 7, 8, 10}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 3, 4, 8, 10}
5
[2, 2, 3, 3, 3]
2.60
d (minimal)
• {1, 3, 4, 5, 7, 10}
6
[2, 2, 3, 3, 3, 4]
2.83
e (minimal)
• {1, 3, 4, 5, 7, 11}
6
[2, 2, 3, 3, 4, 4]
3.00
f (minimal)
• {1, 3, 4, 6, 7, 11}
6
[2, 2, 3, 3, 4, 4]
3.00
g (minimal)
• {1, 2, 4, 5, 7, 11}
6
[2, 2, 3, 4, 4, 4]
3.17
h (minimal)
• {1, 2, 4, 7, 9, 11}
6
[2, 2, 3, 4, 4, 4]
3.17
i (minimal)
• {1, 2, 4, 6, 7, 11}
6
[2, 2, 3, 4, 4, 4]
3.17
j (minimal)
• {1, 2, 4, 6, 9, 11}
6
[2, 2, 4, 4, 4, 4]
3.33
k (minimal)
• {1, 2, 4, 7, 8, 11}
6
[2, 2, 3, 3, 4, 4]
3.00
l (minimal)
• {1, 4, 5, 7, 10, 11}
6
[2, 2, 3, 3, 4, 4]
3.00
m (minimal)
• {1, 4, 7, 9, 10, 11}
6
[2, 2, 3, 3, 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
4
2
0
0
2.75
5
0
3
14
2.86
6
0
10
53
2.99
7
0
0
99
3.09
8
0
0
80
3.16
9
0
0
36
3.21
10
0
0
9
3.24
11
0
0
1
3.27
Total
2
13
292
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,5,6,0],[0,6,7,1],[1,7,8,5],[1,4,6,2],[2,5,8,3],[3,8,8,4],[4,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[6,14,1,7],[7,15,8,18],[13,5,14,6],[1,16,2,15],[8,11,9,12],[12,17,13,18],[4,16,5,17],[2,10,3,11],[9,3,10,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-11,-2)(14,5,-15,-6)(4,15,-5,-16)(13,16,-14,-17)(2,11,-3,-12)(6,7,-1,-8)(17,8,-18,-9)(9,12,-10,-13)(10,3,-7,-4)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,8)(-2,-12,9,-18)(-3,10,12)(-4,-16,13,-10)(-5,14,16)(-6,-8,17,-14)(-7,6,-15,4)(-9,-13,-17)(-11,2)(1,7,3,11)(5,15)
Multiloop annotated with half-edges
11^3_19 annotated with half-edges