$12^{2}_{265}$ - Minimal pinning sets
Pinning sets for 12^2_265 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_265 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 644
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: 3.19456
on average over minimal pinning sets: 2.93725
on average over optimal pinning sets: 2.9125
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 6, 7, 9, 11}
5
[2, 3, 3, 3, 3]
2.80
B (optimal)
• {1, 6, 7, 10, 11}
5
[2, 3, 3, 3, 3]
2.80
C (optimal)
• {1, 6, 9, 10, 11}
5
[2, 3, 3, 3, 3]
2.80
D (optimal)
• {1, 4, 6, 9, 11}
5
[2, 3, 3, 3, 4]
3.00
E (optimal)
• {1, 2, 7, 9, 11}
5
[2, 3, 3, 3, 3]
2.80
F (optimal)
• {1, 2, 6, 7, 10}
5
[2, 3, 3, 3, 3]
2.80
G (optimal)
• {1, 2, 7, 10, 12}
5
[2, 3, 3, 3, 4]
3.00
H (optimal)
• {1, 2, 6, 9, 10}
5
[2, 3, 3, 3, 3]
2.80
I (optimal)
• {1, 2, 7, 10, 11}
5
[2, 3, 3, 3, 3]
2.80
J (optimal)
• {1, 3, 6, 9, 11}
5
[2, 3, 3, 3, 4]
3.00
K (optimal)
• {1, 3, 6, 7, 10}
5
[2, 3, 3, 3, 4]
3.00
L (optimal)
• {1, 3, 6, 9, 10}
5
[2, 3, 3, 3, 4]
3.00
M (optimal)
• {1, 5, 6, 9, 11}
5
[2, 3, 3, 3, 4]
3.00
N (optimal)
• {1, 5, 7, 9, 11}
5
[2, 3, 3, 3, 4]
3.00
O (optimal)
• {1, 5, 7, 10, 11}
5
[2, 3, 3, 3, 4]
3.00
P (optimal)
• {1, 6, 8, 9, 11}
5
[2, 3, 3, 3, 4]
3.00
a (minimal)
• {1, 3, 5, 7, 10, 12}
6
[2, 3, 3, 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
2.91
6
0
1
78
3.05
7
0
0
166
3.15
8
0
0
191
3.22
9
0
0
129
3.27
10
0
0
51
3.3
11
0
0
11
3.32
12
0
0
1
3.33
Total
16
1
627
Other information about this multiloop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,3],[0,2,7,8],[0,8,9,5],[1,4,7,6],[1,5,7,2],[3,6,5,9],[3,9,9,4],[4,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,17,10,16],[7,15,8,16],[19,14,20,15],[1,4,2,5],[17,5,18,6],[10,6,11,7],[11,18,12,19],[3,13,4,14],[2,13,3,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,2,-8,-3)(14,3,-15,-4)(19,4,-20,-5)(9,8,-10,-1)(1,10,-2,-11)(16,11,-17,-12)(5,12,-6,-13)(13,18,-14,-19)(20,15,-9,-16)(6,17,-7,-18)
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)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-11,16,-9)(-2,7,17,11)(-3,14,18,-7)(-4,19,-14)(-5,-13,-19)(-6,-18,13)(-8,9,15,3)(-10,1)(-12,5,-20,-16)(-15,20,4)(-17,6,12)(2,10,8)
Multiloop annotated with half-edges
12^2_265 annotated with half-edges