$12^{2}_{178}$ - Minimal pinning sets
Pinning sets for 12^2_178 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_178 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 240
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.03455
on average over minimal pinning sets: 2.45
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)
• {1, 3, 4, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 4, 5, 8, 9}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 4, 5, 9, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
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
2
13
2.66
7
0
0
45
2.86
8
0
0
71
3.02
9
0
0
64
3.14
10
0
0
33
3.23
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
2
236
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,7,7,8],[0,9,9,5],[1,4,2,1],[2,9,8,7],[3,6,8,3],[3,7,6,9],[4,8,6,4]]
PD code (use to draw this multiloop with SnapPy): [[12,20,1,13],[13,10,14,9],[11,8,12,9],[5,19,6,20],[1,16,2,15],[10,15,11,14],[17,7,18,8],[18,4,19,5],[6,4,7,3],[16,3,17,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,13,-1,-14)(1,4,-2,-5)(15,2,-16,-3)(14,5,-15,-6)(9,6,-10,-7)(7,18,-8,-19)(19,8,-20,-9)(20,11,-13,-12)(3,16,-4,-17)(10,17,-11,-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,-5,14)(-2,15,5)(-3,-17,10,6,-15)(-4,1,13,11,17)(-6,9,-20,-12,-14)(-7,-19,-9)(-8,19)(-10,-18,7)(-11,20,8,18)(-13,12)(-16,3)(2,4,16)
Multiloop annotated with half-edges
12^2_178 annotated with half-edges