$11^{3}_{2}$ - Minimal pinning sets
Pinning sets for 11^3_2 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^3_2 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 139
of which optimal: 2
of which minimal: 9
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98705
on average over minimal pinning sets: 2.66296
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 5, 7, 8}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 5, 8, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
c (minimal)
• {1, 3, 4, 5, 8, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {1, 2, 3, 4, 8, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
e (minimal)
• {1, 2, 3, 5, 8, 9}
6
[2, 2, 2, 3, 3, 4]
2.67
f (minimal)
• {1, 3, 4, 8, 9, 11}
6
[2, 2, 2, 3, 4, 5]
3.00
g (minimal)
• {1, 3, 4, 6, 8, 9}
6
[2, 2, 2, 3, 4, 6]
3.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.4
6
0
7
11
2.7
7
0
0
40
2.91
8
0
0
44
3.05
9
0
0
26
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
2
7
130
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],[0,4,5,5],[0,6,7,3],[0,2,7,7],[0,8,8,1],[1,8,6,1],[2,5,8,7],[2,6,3,3],[4,6,5,4]]
PD code (use to draw this multiloop with SnapPy): [[8,14,1,9],[9,15,10,18],[4,7,5,8],[5,13,6,14],[1,16,2,15],[10,17,11,18],[11,3,12,4],[12,6,13,7],[16,3,17,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,9,-1,-10)(18,1,-13,-2)(13,4,-14,-5)(2,5,-3,-6)(3,14,-4,-15)(6,15,-7,-16)(17,10,-18,-11)(11,16,-12,-17)(12,7,-9,-8)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,10)(-2,-6,-16,11,-18)(-3,-15,6)(-4,13,1,9,7,15)(-5,2,-13)(-7,12,16)(-8,-10,17,-12)(-9,8)(-11,-17)(-14,3,5)(4,14)
Multiloop annotated with half-edges
11^3_2 annotated with half-edges