$12^{2}_{325}$ - Minimal pinning sets
Pinning sets for 12^2_325 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_325 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 680
of which optimal: 2
of which minimal: 14
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.15721
on average over minimal pinning sets: 2.88333
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)
• {1, 6, 9, 11}
4
[2, 2, 4, 4]
3.00
B (optimal)
• {1, 5, 7, 9}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 4, 5, 9, 11}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 3, 4, 9, 11}
5
[2, 2, 3, 4, 4]
3.00
c (minimal)
• {1, 6, 9, 10, 12}
5
[2, 2, 3, 4, 4]
3.00
d (minimal)
• {1, 2, 4, 9, 11}
5
[2, 2, 3, 4, 4]
3.00
e (minimal)
• {1, 3, 7, 9, 11}
5
[2, 2, 3, 4, 4]
3.00
f (minimal)
• {1, 3, 7, 9, 12}
5
[2, 2, 3, 3, 4]
2.80
g (minimal)
• {1, 6, 7, 9, 12}
5
[2, 2, 3, 3, 4]
2.80
h (minimal)
• {1, 4, 5, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
i (minimal)
• {1, 5, 6, 8, 9}
5
[2, 2, 3, 4, 4]
3.00
j (minimal)
• {1, 6, 8, 9, 12}
5
[2, 2, 3, 4, 4]
3.00
k (minimal)
• {1, 4, 5, 9, 10, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
l (minimal)
• {1, 2, 4, 7, 9, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.75
5
0
10
16
2.92
6
0
2
99
3.04
7
0
0
186
3.13
8
0
0
191
3.19
9
0
0
118
3.24
10
0
0
45
3.28
11
0
0
10
3.31
12
0
0
1
3.33
Total
2
12
666
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,8],[0,8,8,9],[0,9,7,5],[1,4,6,1],[1,5,7,2],[2,6,4,9],[2,9,3,3],[3,8,7,4]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,15,6,14],[17,3,18,4],[10,19,11,20],[1,8,2,7],[15,7,16,6],[16,13,17,14],[2,12,3,13],[18,9,19,10],[11,9,12,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(12,15,-13,-16)(6,13,-7,-14)(3,16,-4,-17)(20,17,-5,-18)(18,9,-19,-10)(14,7,-15,-8)(1,8,-2,-9)(10,19,-11,-20)(11,2,-12,-3)
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,-9,18,-5)(-2,11,19,9)(-3,-17,20,-11)(-4,5,17)(-6,-14,-8,1)(-7,14)(-10,-20,-18)(-12,-16,3)(-13,6,4,16)(-15,12,2,8)(-19,10)(7,13,15)
Multiloop annotated with half-edges
12^2_325 annotated with half-edges