$12^{2}_{121}$ - Minimal pinning sets
Pinning sets for 12^2_121 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_121 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, 3, 4, 6, 9}
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, 3, 3, 4, 5, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,7,7],[0,7,7,8],[0,5,1,1],[1,4,9,9],[2,9,8,8],[2,3,3,2],[3,6,6,9],[5,8,6,5]]
PD code (use to draw this multiloop with SnapPy): [[3,12,4,1],[2,20,3,13],[11,6,12,7],[4,10,5,9],[1,14,2,13],[14,19,15,20],[7,18,8,17],[5,10,6,11],[8,16,9,17],[18,15,19,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(4,11,-5,-12)(12,5,-1,-6)(10,7,-11,-8)(14,19,-15,-20)(3,20,-4,-13)(13,2,-14,-3)(18,15,-19,-16)(9,16,-10,-17)(17,8,-18,-9)
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)(-2,13,-4,-12,-6)(-3,-13)(-5,12)(-7,10,16,-19,14,2)(-8,17,-10)(-9,-17)(-11,4,20,-15,18,8)(-14,-20,3)(-16,9,-18)(1,5,11,7)(15,19)
Multiloop annotated with half-edges
12^2_121 annotated with half-edges