$12^{2}_{326}$ - Minimal pinning sets
Pinning sets for 12^2_326 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_326 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 440
of which optimal: 2
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.04903
on average over minimal pinning sets: 2.58
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 7, 9}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 2, 3, 9}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 3, 6, 9, 10}
5
[2, 2, 2, 4, 4]
2.80
b (minimal)
• {1, 3, 6, 8, 9}
5
[2, 2, 2, 4, 4]
2.80
c (minimal)
• {1, 3, 6, 9, 11}
5
[2, 2, 2, 4, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.25
5
0
3
15
2.62
6
0
0
61
2.85
7
0
0
110
3.0
8
0
0
120
3.1
9
0
0
83
3.18
10
0
0
36
3.24
11
0
0
9
3.29
12
0
0
1
3.33
Total
2
3
435
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,4,4,5],[0,5,6,0],[0,6,7,4],[1,3,8,1],[1,8,6,2],[2,5,9,3],[3,9,9,8],[4,7,9,5],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[6,20,1,7],[7,13,8,12],[19,5,20,6],[1,10,2,9],[13,9,14,8],[18,11,19,12],[4,10,5,11],[2,16,3,17],[14,17,15,18],[15,3,16,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(11,2,-12,-3)(14,19,-15,-20)(15,4,-16,-5)(1,16,-2,-17)(10,17,-11,-18)(18,13,-19,-14)(5,8,-6,-9)(20,9,-7,-10)(3,12,-4,-13)
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,-17,10,-7)(-2,11,17)(-3,-13,18,-11)(-4,15,19,13)(-5,-9,20,-15)(-6,7,9)(-8,5,-16,1)(-10,-18,-14,-20)(-12,3)(-19,14)(2,16,4,12)(6,8)
Multiloop annotated with half-edges
12^2_326 annotated with half-edges