$12^{2}_{333}$ - Minimal pinning sets
Pinning sets for 12^2_333 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_333 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 672
of which optimal: 3
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.19344
on average over minimal pinning sets: 2.88333
on average over optimal pinning sets: 2.83333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 5, 9, 12}
4
[2, 3, 3, 3]
2.75
B (optimal)
• {1, 3, 7, 11}
4
[2, 3, 3, 3]
2.75
C (optimal)
• {1, 5, 10, 11}
4
[2, 3, 3, 4]
3.00
a (minimal)
• {1, 3, 7, 9, 12}
5
[2, 3, 3, 3, 3]
2.80
b (minimal)
• {1, 3, 5, 8, 11, 12}
6
[2, 3, 3, 3, 3, 4]
3.00
c (minimal)
• {1, 4, 5, 7, 9, 11}
6
[2, 3, 3, 3, 3, 4]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.83
5
0
1
24
2.98
6
0
2
87
3.08
7
0
0
171
3.16
8
0
0
192
3.22
9
0
0
129
3.27
10
0
0
51
3.3
11
0
0
11
3.32
12
0
0
1
3.33
Total
3
3
666
Other information about this multiloop Properties
Region degree sequence: [2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,3],[0,2,8,4],[0,3,9,1],[1,9,7,6],[1,5,7,2],[2,6,5,8],[3,7,9,9],[4,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[10,5,1,6],[6,11,7,20],[9,13,10,14],[4,12,5,13],[1,12,2,11],[7,16,8,15],[19,14,20,15],[8,18,9,19],[3,17,4,18],[2,17,3,16]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,3,-9,-4)(2,17,-3,-18)(7,18,-8,-19)(14,19,-15,-20)(20,5,-11,-6)(11,10,-12,-1)(1,12,-2,-13)(6,13,-7,-14)(15,4,-16,-5)(16,9,-17,-10)
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,-13,6,-11)(-2,-18,7,13)(-3,8,18)(-4,15,19,-8)(-5,20,-15)(-6,-14,-20)(-7,-19,14)(-9,16,4)(-10,11,5,-16)(-12,1)(-17,2,12,10)(3,17,9)
Multiloop annotated with half-edges
12^2_333 annotated with half-edges