$10^{1}_{18}$ - Minimal pinning sets
Pinning sets for 10^1_18 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_18 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 160
of which optimal: 10
of which minimal: 10
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.14988
on average over minimal pinning sets: 3.04
on average over optimal pinning sets: 3.04
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 7, 9}
5
[3, 3, 3, 3, 3]
3.00
B (optimal)
• {1, 3, 6, 8, 9}
5
[3, 3, 3, 3, 4]
3.20
C (optimal)
• {1, 2, 5, 6, 9}
5
[3, 3, 3, 3, 3]
3.00
D (optimal)
• {1, 2, 5, 7, 9}
5
[3, 3, 3, 3, 3]
3.00
E (optimal)
• {2, 3, 6, 7, 10}
5
[3, 3, 3, 3, 3]
3.00
F (optimal)
• {2, 4, 5, 7, 10}
5
[3, 3, 3, 3, 4]
3.20
G (optimal)
• {1, 3, 6, 7, 10}
5
[3, 3, 3, 3, 3]
3.00
H (optimal)
• {1, 3, 5, 7, 10}
5
[3, 3, 3, 3, 3]
3.00
I (optimal)
• {2, 5, 6, 9, 10}
5
[3, 3, 3, 3, 3]
3.00
J (optimal)
• {2, 3, 6, 9, 10}
5
[3, 3, 3, 3, 3]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
10
0
0
3.04
6
0
0
42
3.11
7
0
0
60
3.16
8
0
0
37
3.19
9
0
0
10
3.2
10
0
0
1
3.2
Total
10
0
150
Other information about this loop Properties
Region degree sequence: [3, 3, 3, 3, 3, 3, 3, 3, 4, 4]
Minimal region degree: 3
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,3],[0,2,7,4],[0,3,7,5],[1,4,7,6],[1,5,7,2],[3,6,5,4]]
PD code (use to draw this loop with SnapPy): [[5,16,6,1],[9,4,10,5],[10,15,11,16],[6,11,7,12],[1,12,2,13],[13,8,14,9],[14,3,15,4],[7,3,8,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,5,-1,-6)(12,1,-13,-2)(7,2,-8,-3)(11,6,-12,-7)(4,9,-5,-10)(15,10,-16,-11)(8,13,-9,-14)(3,14,-4,-15)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,6)(-2,7,-12)(-3,-15,-11,-7)(-4,-10,15)(-5,16,10)(-6,11,-16)(-8,-14,3)(-9,4,14)(-13,8,2)(1,5,9,13)
Loop annotated with half-edges
10^1_18 annotated with half-edges