$12^{2}_{301}$ - Minimal pinning sets
Pinning sets for 12^2_301 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_301 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 767
of which optimal: 12
of which minimal: 24
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.27221
on average over minimal pinning sets: 3.15
on average over optimal pinning sets: 3.13333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {3, 6, 8, 10, 11}
5
[3, 3, 3, 3, 4]
3.20
B (optimal)
• {3, 6, 7, 10, 11}
5
[3, 3, 3, 3, 3]
3.00
C (optimal)
• {3, 4, 8, 10, 11}
5
[3, 3, 3, 3, 4]
3.20
D (optimal)
• {3, 4, 7, 9, 12}
5
[3, 3, 3, 3, 4]
3.20
E (optimal)
• {2, 3, 6, 10, 11}
5
[3, 3, 3, 3, 4]
3.20
F (optimal)
• {2, 3, 6, 9, 10}
5
[3, 3, 3, 3, 4]
3.20
G (optimal)
• {1, 4, 7, 9, 11}
5
[3, 3, 3, 3, 3]
3.00
H (optimal)
• {1, 4, 7, 9, 12}
5
[3, 3, 3, 3, 4]
3.20
I (optimal)
• {1, 4, 6, 7, 9}
5
[3, 3, 3, 3, 3]
3.00
J (optimal)
• {1, 4, 5, 9, 10}
5
[3, 3, 3, 3, 4]
3.20
K (optimal)
• {1, 4, 5, 7, 9}
5
[3, 3, 3, 3, 4]
3.20
L (optimal)
• {1, 3, 6, 10, 11}
5
[3, 3, 3, 3, 3]
3.00
a (minimal)
• {3, 6, 7, 9, 11, 12}
6
[3, 3, 3, 3, 3, 4]
3.17
b (minimal)
• {3, 4, 8, 9, 10, 12}
6
[3, 3, 3, 3, 4, 4]
3.33
c (minimal)
• {3, 4, 7, 9, 10, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
d (minimal)
• {2, 3, 6, 7, 9, 12}
6
[3, 3, 3, 3, 4, 4]
3.33
e (minimal)
• {2, 3, 4, 5, 9, 10}
6
[3, 3, 3, 3, 4, 4]
3.33
f (minimal)
• {1, 4, 7, 8, 10, 11}
6
[3, 3, 3, 3, 3, 4]
3.17
g (minimal)
• {1, 4, 6, 7, 10, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
h (minimal)
• {1, 4, 5, 8, 10, 11}
6
[3, 3, 3, 3, 4, 4]
3.33
i (minimal)
• {1, 4, 5, 6, 10, 11}
6
[3, 3, 3, 3, 3, 4]
3.17
j (minimal)
• {1, 3, 6, 7, 9, 11}
6
[3, 3, 3, 3, 3, 3]
3.00
k (minimal)
• {1, 3, 4, 6, 9, 10}
6
[3, 3, 3, 3, 3, 3]
3.00
l (minimal)
• {1, 2, 3, 6, 7, 9}
6
[3, 3, 3, 3, 3, 4]
3.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
12
0
0
3.13
6
0
12
68
3.19
7
0
0
200
3.25
8
0
0
242
3.29
9
0
0
160
3.31
10
0
0
60
3.33
11
0
0
12
3.33
12
0
0
1
3.33
Total
12
12
743
Other information about this multiloop Properties
Region degree sequence: [3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4]
Minimal region degree: 3
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,6,2],[0,1,6,7],[0,7,8,4],[0,3,8,5],[1,4,9,6],[1,5,9,2],[2,9,8,3],[3,7,9,4],[5,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,6,16,5],[13,4,14,5],[9,19,10,20],[1,10,2,11],[6,11,7,12],[16,12,17,13],[8,3,9,4],[18,2,19,3],[7,18,8,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,14,-16,-1)(6,1,-7,-2)(11,2,-12,-3)(3,18,-4,-19)(13,8,-14,-9)(4,9,-5,-10)(19,10,-20,-11)(20,5,-15,-6)(7,16,-8,-17)(12,17,-13,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-15)(-2,11,-20,-6)(-3,-19,-11)(-4,-10,19)(-5,20,10)(-7,-17,12,2)(-8,13,17)(-9,4,18,-13)(-12,-18,3)(-14,15,5,9)(-16,7,1)(8,16,14)
Multiloop annotated with half-edges
12^2_301 annotated with half-edges