$12^{2}_{315}$ - Minimal pinning sets
Pinning sets for 12^2_315 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_315 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 322
of which optimal: 2
of which minimal: 12
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05856
on average over minimal pinning sets: 2.70556
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 7, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {2, 3, 4, 6, 7, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
b (minimal)
• {2, 4, 6, 7, 11, 12}
6
[2, 2, 2, 3, 4, 4]
2.83
c (minimal)
• {1, 4, 5, 7, 8, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
d (minimal)
• {1, 4, 7, 8, 11, 12}
6
[2, 2, 2, 3, 3, 4]
2.67
e (minimal)
• {1, 4, 6, 7, 11, 12}
6
[2, 2, 2, 3, 4, 4]
2.83
f (minimal)
• {1, 4, 5, 7, 11, 12}
6
[2, 2, 2, 3, 4, 4]
2.83
g (minimal)
• {2, 4, 6, 7, 9, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
h (minimal)
• {2, 4, 5, 7, 9, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
i (minimal)
• {2, 3, 4, 7, 9, 11}
6
[2, 2, 2, 3, 3, 4]
2.67
j (minimal)
• {1, 4, 5, 7, 9, 11}
6
[2, 2, 2, 3, 4, 4]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.4
6
0
10
14
2.71
7
0
0
73
2.93
8
0
0
101
3.08
9
0
0
77
3.17
10
0
0
35
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
10
310
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,3],[0,2,8,4],[0,3,9,5],[1,4,9,1],[1,7,7,2],[2,6,6,8],[3,7,9,9],[4,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[14,5,1,6],[6,15,7,20],[13,10,14,11],[4,9,5,10],[1,9,2,8],[15,8,16,7],[19,11,20,12],[12,18,13,19],[3,17,4,18],[2,17,3,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,3,-13,-4)(5,20,-6,-15)(6,13,-7,-14)(2,7,-3,-8)(11,8,-12,-9)(18,9,-19,-10)(15,14,-16,-1)(1,16,-2,-17)(10,17,-11,-18)(19,4,-20,-5)
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,-17,10,-19,-5,-15)(-2,-8,11,17)(-3,12,8)(-4,19,9,-12)(-6,-14,15)(-7,2,16,14)(-9,18,-11)(-10,-18)(-13,6,20,4)(-16,1)(-20,5)(3,7,13)
Multiloop annotated with half-edges
12^2_315 annotated with half-edges