$12^{3}_{2}$ - Minimal pinning sets
Pinning sets for 12^3_2 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_2 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 445
of which optimal: 2
of which minimal: 7
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04925
on average over minimal pinning sets: 2.68095
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 6, 8, 10}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 6, 10, 11}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 5, 6, 10, 12}
5
[2, 2, 2, 4, 4]
2.80
b (minimal)
• {1, 3, 6, 10, 12}
5
[2, 2, 2, 4, 4]
2.80
c (minimal)
• {1, 6, 7, 10, 12}
5
[2, 2, 2, 4, 5]
3.00
d (minimal)
• {1, 6, 9, 10, 12}
5
[2, 2, 2, 4, 5]
3.00
e (minimal)
• {1, 2, 4, 6, 10, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
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
2.25
5
0
4
15
2.65
6
0
1
63
2.85
7
0
0
111
3.0
8
0
0
120
3.1
9
0
0
83
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
5
438
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,5],[0,5,4,3],[0,2,1,0],[1,2,6,7],[1,8,9,2],[4,9,9,7],[4,6,8,8],[5,7,7,9],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,15,10,14],[7,2,8,3],[15,1,16,2],[10,7,11,6],[13,3,14,4],[11,17,12,20],[5,19,6,20],[4,19,5,18],[12,17,13,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(3,14,-4,-15)(11,6,-12,-7)(4,7,-5,-8)(10,13,-11,-14)(5,12,-6,-13)(20,1,-17,-2)(2,17,-3,-18)(18,15,-19,-16)(16,19,-9,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,-9)(-2,-18,-16,-20)(-3,-15,18)(-4,-8,9,19,15)(-5,-13,10,8)(-6,11,13)(-7,4,14,-11)(-10,-14,3,17,1)(-12,5,7)(-17,2)(-19,16)(6,12)
Multiloop annotated with half-edges
12^3_2 annotated with half-edges