$12^{2}_{45}$ - Minimal pinning sets
Pinning sets for 12^2_45
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_45
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, 7, 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, 4, 4, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,5],[0,6,7,7],[0,7,7,8],[0,9,9,1],[1,9,6,1],[2,5,8,8],[2,3,3,2],[3,6,6,9],[4,8,5,4]]
PD code (use to draw this multiloop with SnapPy): [[7,16,8,1],[6,20,7,17],[15,10,16,11],[8,14,9,13],[1,18,2,17],[19,5,20,6],[11,5,12,4],[9,14,10,15],[12,3,13,4],[18,3,19,2]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,17,-1,-18)(10,1,-11,-2)(2,9,-3,-10)(18,3,-19,-4)(13,8,-14,-9)(11,6,-12,-7)(7,12,-8,-13)(5,14,-6,-15)(4,19,-5,-20)(20,15,-17,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10,-3,18)(-2,-10)(-4,-20,-16,-18)(-5,-15,20)(-6,11,1,17,15)(-7,-13,-9,2,-11)(-8,13)(-12,7)(-14,5,19,3,9)(-17,16)(-19,4)(6,14,8,12)
Multiloop annotated with half-edges
12^2_45 annotated with half-edges