$10^{2}_{27}$ - Minimal pinning sets
Pinning sets for 10^2_27 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_27 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 153
of which optimal: 16
of which minimal: 17
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.02568
on average over minimal pinning sets: 2.83137
on average over optimal pinning sets: 2.8
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 7, 9}
5
[2, 2, 3, 3, 4]
2.80
B (optimal)
• {2, 3, 7, 9, 10}
5
[2, 2, 3, 4, 4]
3.00
C (optimal)
• {2, 3, 4, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
D (optimal)
• {1, 2, 3, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
E (optimal)
• {2, 3, 7, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
F (optimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 3, 3, 3]
2.60
G (optimal)
• {1, 3, 5, 9, 10}
5
[2, 2, 3, 3, 4]
2.80
H (optimal)
• {3, 5, 7, 9, 10}
5
[2, 2, 3, 4, 4]
3.00
I (optimal)
• {2, 3, 5, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
J (optimal)
• {3, 4, 5, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
K (optimal)
• {1, 3, 5, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
L (optimal)
• {3, 5, 8, 9, 10}
5
[2, 2, 3, 3, 4]
2.80
M (optimal)
• {1, 2, 3, 6, 9}
5
[2, 2, 3, 3, 4]
2.80
N (optimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 3, 4, 4]
3.00
O (optimal)
• {3, 4, 6, 8, 9}
5
[2, 2, 3, 4, 4]
3.00
P (optimal)
• {1, 3, 5, 6, 9}
5
[2, 2, 3, 3, 4]
2.80
a (minimal)
• {3, 4, 6, 7, 9, 10}
6
[2, 2, 4, 4, 4, 4]
3.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
16
0
0
2.8
6
0
1
47
2.97
7
0
0
52
3.07
8
0
0
28
3.12
9
0
0
8
3.17
10
0
0
1
3.2
Total
16
1
136
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,5,6,3],[0,2,7,7],[0,7,6,1],[1,6,2,1],[2,5,4,7],[3,6,4,3]]
PD code (use to draw this multiloop with SnapPy): [[4,16,1,5],[5,13,6,12],[7,3,8,4],[8,15,9,16],[1,14,2,13],[6,11,7,12],[2,10,3,11],[14,9,15,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(9,16,-10,-5)(4,5,-1,-6)(13,6,-14,-7)(7,12,-8,-13)(8,3,-9,-4)(15,10,-16,-11)(2,11,-3,-12)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,6)(-2,-12,7,-14)(-3,8,12)(-4,-6,13,-8)(-5,4,-9)(-7,-13)(-10,15,1,5)(-11,2,-15)(-16,9,3,11)(10,16)
Multiloop annotated with half-edges
10^2_27 annotated with half-edges