$12^{2}_{245}$ - Minimal pinning sets
Pinning sets for 12^2_245 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_245 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 192
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.96906
on average over minimal pinning sets: 2.2
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 5, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
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
2.2
6
0
0
13
2.54
7
0
0
36
2.78
8
0
0
55
2.95
9
0
0
50
3.09
10
0
0
27
3.19
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
0
190
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,7],[0,8,8,4],[0,3,8,5],[1,4,6,1],[2,5,9,9],[2,9,9,8],[3,7,4,3],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[7,14,8,1],[6,20,7,15],[13,19,14,20],[8,3,9,4],[1,4,2,5],[15,5,16,6],[16,12,17,13],[18,10,19,11],[2,9,3,10],[11,17,12,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(13,2,-14,-3)(16,3,-17,-4)(14,7,-1,-8)(19,8,-20,-9)(17,10,-18,-11)(5,12,-6,-13)(9,18,-10,-19)(11,20,-12,-15)(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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,12,20,8)(-2,13,-6)(-3,16,-5,-13)(-4,-16)(-7,14,2)(-8,19,-10,17,3,-14)(-9,-19)(-11,-15,4,-17)(-12,5,15)(-18,9,-20,11)(1,7)(10,18)
Multiloop annotated with half-edges
12^2_245 annotated with half-edges