$12^{2}_{8}$ - Minimal pinning sets
Pinning sets for 12^2_8 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_8 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, 3, 4, 7}
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, 4, 4, 5, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,6,6,7],[0,7,4,4],[1,3,3,7],[1,8,8,9],[2,9,9,2],[2,8,4,3],[5,7,9,5],[5,8,6,6]]
PD code (use to draw this multiloop with SnapPy): [[4,20,1,5],[5,3,6,4],[19,12,20,13],[1,9,2,10],[10,2,11,3],[6,16,7,15],[13,18,14,19],[8,11,9,12],[16,8,17,7],[17,14,18,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(16,1,-17,-2)(18,11,-19,-12)(3,20,-4,-5)(19,6,-20,-7)(14,7,-15,-8)(12,9,-13,-10)(10,17,-11,-18)(8,13,-9,-14)(2,15,-3,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-3,-5)(-2,-16)(-4,5)(-6,19,11,17,1)(-7,14,-9,12,-19)(-8,-14)(-10,-18,-12)(-11,18)(-13,8,-15,2,-17,10)(-20,3,15,7)(4,20,6)(9,13)
Multiloop annotated with half-edges
12^2_8 annotated with half-edges