$12^{2}_{50}$ - Minimal pinning sets
Pinning sets for 12^2_50 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_50 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 446
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: 3.0488
on average over minimal pinning sets: 2.53571
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 7, 11}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {2, 4, 7, 11}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {4, 5, 7, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {4, 5, 7, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
• {4, 5, 6, 7, 11}
5
[2, 2, 2, 3, 6]
3.00
d (minimal)
• {4, 5, 7, 11, 12}
5
[2, 2, 2, 3, 4]
2.60
e (minimal)
• {3, 4, 5, 7, 11}
5
[2, 2, 2, 3, 3]
2.40
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
2.38
5
0
5
15
2.65
6
0
0
64
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
439
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],[0,4,4,5],[0,5,6,3],[0,2,7,8],[0,5,1,1],[1,4,9,2],[2,9,9,7],[3,6,8,8],[3,7,7,9],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[3,16,4,1],[11,2,12,3],[15,8,16,9],[4,8,5,7],[1,10,2,11],[12,10,13,9],[14,20,15,17],[5,20,6,19],[6,18,7,19],[13,18,14,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,16,-10,-1)(5,2,-6,-3)(13,6,-14,-7)(7,10,-8,-11)(15,8,-16,-9)(1,14,-2,-15)(18,3,-19,-4)(12,19,-13,-20)(20,11,-17,-12)(4,17,-5,-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,-15,-9)(-2,5,17,11,-8,15)(-3,18,-5)(-4,-18)(-6,13,19,3)(-7,-11,20,-13)(-10,7,-14,1)(-12,-20)(-16,9)(-17,4,-19,12)(2,14,6)(8,10,16)
Multiloop annotated with half-edges
12^2_50 annotated with half-edges