$12^{4}_{2}$ - Minimal pinning sets
Pinning sets for 12^4_2 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^4_2 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, 2, 5, 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, 3, 3, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,5],[0,6,7,7],[0,7,4,0],[1,3,5,1],[1,4,8,9],[2,9,8,7],[2,6,3,2],[5,6,9,9],[5,8,8,6]]
PD code (use to draw this multiloop with SnapPy): [[4,8,1,5],[5,9,6,14],[3,20,4,15],[7,1,8,2],[9,7,10,6],[10,13,11,14],[15,18,16,19],[19,2,20,3],[16,12,17,13],[11,17,12,18]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-5,-2)(15,2,-16,-3)(7,10,-8,-11)(4,5,-1,-6)(18,13,-19,-14)(11,14,-12,-9)(9,6,-10,-7)(17,20,-18,-15)(3,16,-4,-17)(12,19,-13,-20)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,10,6)(-2,15,-18,-14,11,-8)(-3,-17,-15)(-4,-6,9,-12,-20,17)(-5,4,16,2)(-7,-11,-9)(-10,7)(-13,18,20)(-16,3)(-19,12,14)(1,5)(13,19)
Multiloop annotated with half-edges
12^4_2 annotated with half-edges