$11^{3}_{13}$ - Minimal pinning sets
Pinning sets for 11^3_13 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^3_13 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 183
of which optimal: 1
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97971
on average over minimal pinning sets: 2.65278
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 8}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 4, 5, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 4, 5, 7, 8}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
• {1, 4, 5, 6, 8, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
d (minimal)
• {1, 4, 5, 8, 9, 11}
6
[2, 2, 2, 3, 4, 5]
3.00
e (minimal)
• {1, 2, 4, 5, 8, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
2
7
2.56
6
0
3
30
2.79
7
0
0
54
2.96
8
0
0
50
3.08
9
0
0
27
3.16
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
5
177
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],[0,5,6,7],[0,7,7,3],[0,2,8,8],[0,6,5,5],[1,4,4,6],[1,5,4,8],[1,8,2,2],[3,7,6,3]]
PD code (use to draw this multiloop with SnapPy): [[6,14,1,7],[7,3,8,4],[5,18,6,15],[13,17,14,18],[1,9,2,10],[10,2,11,3],[8,11,9,12],[4,16,5,15],[16,12,17,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(12,1,-13,-2)(5,14,-6,-7)(13,8,-14,-9)(16,9,-17,-10)(4,11,-5,-12)(10,17,-11,-18)(3,18,-4,-15)(15,2,-16,-3)
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,-5,-7)(-2,15,-4,-12)(-3,-15)(-6,7)(-8,13,1)(-9,16,2,-13)(-10,-18,3,-16)(-11,4,18)(-14,5,11,17,9)(-17,10)(6,14,8)
Multiloop annotated with half-edges
11^3_13 annotated with half-edges