$12^{2}_{156}$ - Minimal pinning sets
Pinning sets for 12^2_156 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_156 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90623
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,7,7],[0,8,4,4],[0,3,3,5],[1,4,9,1],[1,9,9,2],[2,8,8,2],[3,7,7,9],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[16,3,1,4],[4,17,5,20],[15,10,16,11],[2,7,3,8],[1,7,2,6],[17,6,18,5],[19,11,20,12],[9,14,10,15],[8,14,9,13],[18,13,19,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(3,18,-4,-19)(4,15,-5,-16)(16,5,-1,-6)(13,8,-14,-9)(20,9,-17,-10)(10,19,-11,-20)(11,6,-12,-7)(7,12,-8,-13)(17,2,-18,-3)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14,8,12,6)(-2,17,9,-14)(-3,-19,10,-17)(-4,-16,-6,11,19)(-5,16)(-7,-13,-9,20,-11)(-8,13)(-10,-20)(-12,7)(-15,4,18,2)(-18,3)(1,5,15)
Multiloop annotated with half-edges
12^2_156 annotated with half-edges