$12^{2}_{330}$ - Minimal pinning sets
Pinning sets for 12^2_330 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_330 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 560
of which optimal: 1
of which minimal: 11
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.13025
on average over minimal pinning sets: 2.80606
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 12}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 5, 12}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 2, 4, 11, 12}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {1, 3, 4, 6, 10}
5
[2, 2, 3, 4, 4]
3.00
d (minimal)
• {1, 2, 4, 9, 12}
5
[2, 2, 3, 3, 3]
2.60
e (minimal)
• {1, 2, 4, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
f (minimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 3, 3, 4]
2.80
g (minimal)
• {1, 3, 4, 7, 10}
5
[2, 2, 3, 3, 4]
2.80
h (minimal)
• {1, 3, 4, 7, 9}
5
[2, 2, 3, 3, 3]
2.60
i (minimal)
• {1, 2, 4, 8, 12}
5
[2, 2, 3, 3, 5]
3.00
j (minimal)
• {1, 2, 4, 6, 10, 11}
6
[2, 2, 3, 4, 4, 4]
3.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.5
5
0
9
8
2.78
6
0
1
71
2.96
7
0
0
144
3.08
8
0
0
162
3.17
9
0
0
109
3.23
10
0
0
44
3.28
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
10
549
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,3],[0,2,5,8],[0,8,8,1],[1,8,3,9],[1,9,7,2],[2,6,9,9],[3,5,4,4],[5,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[16,20,1,17],[17,10,18,9],[15,4,16,5],[19,3,20,4],[1,11,2,10],[18,12,19,13],[8,5,9,6],[14,7,15,8],[2,11,3,12],[13,7,14,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (18,1,-19,-2)(13,2,-14,-3)(6,3,-7,-4)(4,11,-5,-12)(12,5,-13,-6)(7,20,-8,-17)(8,15,-9,-16)(16,9,-1,-10)(17,10,-18,-11)(14,19,-15,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,18,10)(-2,13,5,11,-18)(-3,6,-13)(-4,-12,-6)(-5,12)(-7,-17,-11,4)(-8,-16,-10,17)(-9,16)(-14,-20,7,3)(-15,8,20)(-19,14,2)(1,9,15,19)
Multiloop annotated with half-edges
12^2_330 annotated with half-edges