$12^{2}_{65}$ - Minimal pinning sets
Pinning sets for 12^2_65 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_65 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, 2, 3, 5, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 2, 3, 4, 6, 10}
6
[2, 2, 2, 3, 3, 3]
2.50
b (minimal)
• {1, 2, 3, 4, 6, 9}
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: [[1,2,2,3],[0,3,4,2],[0,1,4,0],[0,5,6,1],[1,7,5,2],[3,4,8,6],[3,5,8,7],[4,6,9,9],[5,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[8,20,1,9],[9,18,10,19],[19,7,20,8],[1,17,2,18],[10,6,11,7],[11,16,12,17],[2,12,3,13],[13,5,14,6],[15,3,16,4],[4,14,5,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-9,-2)(17,2,-18,-3)(14,3,-15,-4)(10,7,-11,-8)(6,19,-7,-20)(8,9,-1,-10)(18,11,-19,-12)(5,12,-6,-13)(16,13,-17,-14)(4,15,-5,-16)
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,-7,10)(-2,17,13,-6,-20)(-3,14,-17)(-4,-16,-14)(-5,-13,16)(-8,-10)(-9,8,-11,18,2)(-12,5,15,3,-18)(-15,4)(-19,6,12)(1,9)(7,19,11)
Multiloop annotated with half-edges
12^2_65 annotated with half-edges