$12^{3}_{67}$ - Minimal pinning sets
Pinning sets for 12^3_67 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_67 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 336
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.10243
on average over minimal pinning sets: 2.625
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 6, 7}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 5, 6, 7}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 4, 5, 6, 7, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 3, 4, 6, 7, 11}
6
[2, 2, 3, 3, 3, 3]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.5
5
0
0
8
2.75
6
0
3
28
2.89
7
0
0
70
3.01
8
0
0
96
3.11
9
0
0
80
3.19
10
0
0
39
3.26
11
0
0
10
3.31
12
0
0
1
3.33
Total
1
3
332
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,7],[0,8,5,1],[1,4,8,2],[3,8,9,9],[3,9,9,8],[4,7,6,5],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[4,12,1,5],[5,13,6,20],[3,19,4,20],[11,18,12,19],[1,14,2,13],[6,2,7,3],[15,10,16,11],[17,8,18,9],[14,8,15,7],[9,16,10,17]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(6,15,-7,-16)(18,7,-19,-8)(16,9,-17,-10)(2,11,-3,-12)(8,17,-9,-18)(10,19,-11,-20)(1,20,-2,-13)(13,12,-14,-5)(14,3,-15,-4)
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,-5)(-2,-12,13)(-3,14,12)(-4,5,-14)(-6,-16,-10,-20,1)(-7,18,-9,16)(-8,-18)(-11,2,20)(-15,6,4)(-17,8,-19,10)(3,11,19,7,15)(9,17)
Multiloop annotated with half-edges
12^3_67 annotated with half-edges