$12^{2}_{293}$ - Minimal pinning sets
Pinning sets for 12^2_293 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_293 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 368
of which optimal: 5
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10482
on average over minimal pinning sets: 2.63889
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 2, 3, 5, 7}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 2, 6, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 2, 5, 9, 11}
5
[2, 2, 3, 3, 3]
2.60
E (optimal)
• {1, 2, 5, 7, 11}
5
[2, 2, 3, 3, 3]
2.60
a (minimal)
• {1, 2, 3, 6, 7, 12}
6
[2, 2, 3, 3, 3, 4]
2.83
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
5
0
0
2.6
6
0
1
30
2.83
7
0
0
80
3.0
8
0
0
111
3.12
9
0
0
89
3.21
10
0
0
41
3.27
11
0
0
10
3.31
12
0
0
1
3.33
Total
5
1
362
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,2],[0,1,5,6],[0,7,4,0],[1,3,8,5],[1,4,8,2],[2,8,9,7],[3,6,9,9],[4,9,6,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[7,12,8,1],[6,20,7,13],[11,19,12,20],[8,2,9,1],[13,9,14,10],[10,5,11,6],[15,18,16,19],[2,16,3,17],[14,4,15,5],[17,3,18,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,1,-15,-2)(19,2,-20,-3)(8,3,-9,-4)(18,7,-19,-8)(5,10,-6,-11)(11,6,-12,-7)(12,13,-1,-14)(20,15,-13,-16)(9,16,-10,-17)(4,17,-5,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,14)(-2,19,7,-12,-14)(-3,8,-19)(-4,-18,-8)(-5,-11,-7,18)(-6,11)(-9,-17,4)(-10,5,17)(-13,12,6,10,16)(-15,20,2)(-16,9,3,-20)(1,13,15)
Multiloop annotated with half-edges
12^2_293 annotated with half-edges