$12^{2}_{42}$ - Minimal pinning sets
Pinning sets for 12^2_42 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_42 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91189
on average over minimal pinning sets: 2.16667
on average over optimal pinning sets: 2.16667
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 5, 6, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
B (optimal)
• {1, 3, 4, 5, 6, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
2
0
0
2.17
7
0
0
11
2.52
8
0
0
25
2.78
9
0
0
30
2.98
10
0
0
20
3.13
11
0
0
7
3.25
12
0
0
1
3.33
Total
2
0
94
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],[0,8,8,4],[1,3,8,7],[1,7,9,9],[2,9,9,2],[2,5,4,8],[3,7,4,3],[5,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[14,20,1,15],[15,13,16,14],[7,19,8,20],[1,11,2,10],[12,9,13,10],[16,4,17,5],[18,6,19,7],[8,4,9,3],[11,3,12,2],[17,6,18,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,5,-1,-6)(20,1,-15,-2)(2,15,-3,-16)(16,3,-17,-4)(9,6,-10,-7)(7,12,-8,-13)(19,10,-20,-11)(13,8,-14,-9)(4,17,-5,-18)(11,18,-12,-19)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,10,6)(-2,-16,-4,-18,11,-20)(-3,16)(-5,14,8,12,18)(-6,9,-14)(-7,-13,-9)(-8,13)(-10,19,-12,7)(-11,-19)(-15,2)(-17,4)(1,5,17,3,15)
Multiloop annotated with half-edges
12^2_42 annotated with half-edges