$12^{4}_{8}$ - Minimal pinning sets
Pinning sets for 12^4_8 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_8 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 256
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: 2.96564
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, 4, 6, 9}
4
[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
4
1
0
0
2.0
5
0
0
8
2.4
6
0
0
28
2.67
7
0
0
56
2.86
8
0
0
70
3.0
9
0
0
56
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
0
255
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,5,0],[0,5,5,3],[0,2,6,4],[1,3,7,8],[1,9,2,2],[3,9,7,7],[4,6,6,8],[4,7,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[6,10,1,7],[7,5,8,6],[9,16,10,11],[1,16,2,15],[4,14,5,15],[8,12,9,11],[2,17,3,20],[3,19,4,20],[13,18,14,19],[12,18,13,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (17,2,-18,-3)(3,20,-4,-17)(1,16,-2,-11)(11,4,-12,-5)(19,12,-20,-13)(13,18,-14,-19)(15,10,-16,-7)(6,7,-1,-8)(8,5,-9,-6)(9,14,-10,-15)
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,-11,-5,8)(-2,17,-4,11)(-3,-17)(-6,-8)(-7,6,-9,-15)(-10,15)(-12,19,-14,9,5)(-13,-19)(-16,1,7)(-18,13,-20,3)(2,16,10,14,18)(4,20,12)
Multiloop annotated with half-edges
12^4_8 annotated with half-edges