$12^{2}_{48}$ - Minimal pinning sets
Pinning sets for 12^2_48 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_48 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 100
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9314
on average over minimal pinning sets: 2.29524
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 8, 11}
6
[2, 2, 2, 2, 2, 4]
2.33
a (minimal)
• {1, 2, 3, 4, 5, 6, 11}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
b (minimal)
• {1, 2, 3, 4, 5, 9, 11}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
c (minimal)
• {1, 2, 3, 4, 6, 7, 11}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
d (minimal)
• {1, 2, 3, 4, 7, 9, 11}
7
[2, 2, 2, 2, 2, 3, 3]
2.29
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
1
0
0
2.33
7
0
4
6
2.49
8
0
0
27
2.78
9
0
0
33
3.0
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
4
95
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 6, 8]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,3],[0,2,7,7],[0,5,1,1],[1,4,6,2],[2,5,8,8],[3,9,9,3],[6,9,9,6],[7,8,8,7]]
PD code (use to draw this multiloop with SnapPy): [[3,10,4,1],[2,20,3,11],[13,9,14,10],[4,14,5,15],[1,12,2,11],[12,19,13,20],[8,18,9,19],[5,16,6,15],[17,7,18,8],[16,7,17,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,11,-1,-12)(12,1,-13,-2)(18,3,-19,-4)(7,4,-8,-5)(8,19,-9,-20)(20,9,-11,-10)(2,13,-3,-14)(17,14,-18,-15)(6,15,-7,-16)(16,5,-17,-6)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,-14,17,5,-8,-20,-10,-12)(-3,18,14)(-4,7,15,-18)(-5,16,-7)(-6,-16)(-9,20)(-11,10)(-13,2)(-15,6,-17)(-19,8,4)(1,11,9,19,3,13)
Multiloop annotated with half-edges
12^2_48 annotated with half-edges