$11^{3}_{17}$ - Minimal pinning sets
Pinning sets for 11^3_17 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^3_17 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 257
of which optimal: 1
of which minimal: 13
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.13036
on average over minimal pinning sets: 2.95256
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)
• {2, 3, 6, 8}
4
[2, 3, 3, 3]
2.75
a (minimal)
• {2, 3, 6, 9, 10}
5
[2, 3, 3, 3, 4]
3.00
b (minimal)
• {1, 4, 5, 6, 9}
5
[2, 3, 3, 3, 3]
2.80
c (minimal)
• {1, 2, 4, 5, 6, 8}
6
[2, 3, 3, 3, 3, 3]
2.83
d (minimal)
• {1, 4, 5, 6, 8, 11}
6
[2, 3, 3, 3, 3, 4]
3.00
e (minimal)
• {1, 3, 4, 6, 9, 11}
6
[2, 3, 3, 3, 3, 4]
3.00
f (minimal)
• {1, 2, 3, 6, 9, 11}
6
[2, 3, 3, 3, 3, 4]
3.00
g (minimal)
• {1, 3, 4, 6, 7, 9}
6
[2, 3, 3, 3, 3, 5]
3.17
h (minimal)
• {1, 2, 3, 6, 7, 9}
6
[2, 3, 3, 3, 3, 5]
3.17
i (minimal)
• {1, 3, 4, 6, 8, 11}
6
[2, 3, 3, 3, 3, 4]
3.00
j (minimal)
• {1, 3, 4, 6, 8, 9}
6
[2, 3, 3, 3, 3, 3]
2.83
k (minimal)
• {1, 2, 3, 5, 6, 9}
6
[2, 3, 3, 3, 3, 3]
2.83
l (minimal)
• {1, 3, 4, 6, 9, 10}
6
[2, 3, 3, 3, 3, 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
1
0
0
2.75
5
0
2
7
2.91
6
0
10
32
3.01
7
0
0
79
3.11
8
0
0
76
3.18
9
0
0
39
3.23
10
0
0
10
3.26
11
0
0
1
3.27
Total
1
12
244
Other information about this multiloop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,6],[0,6,7,7],[0,8,5,1],[1,4,8,2],[2,8,7,3],[3,6,8,3],[4,7,6,5]]
PD code (use to draw this multiloop with SnapPy): [[4,10,1,5],[5,11,6,18],[3,17,4,18],[14,9,15,10],[1,12,2,11],[6,2,7,3],[13,16,14,17],[8,15,9,16],[12,8,13,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(6,13,-7,-14)(15,8,-16,-9)(2,9,-3,-10)(7,16,-8,-17)(14,17,-15,-18)(1,18,-2,-11)(11,10,-12,-5)(12,3,-13,-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,-11,-5)(-2,-10,11)(-3,12,10)(-4,5,-12)(-6,-14,-18,1)(-7,-17,14)(-8,15,17)(-9,2,18,-15)(-13,6,4)(-16,7,13,3,9)(8,16)
Multiloop annotated with half-edges
11^3_17 annotated with half-edges