$12^{3}_{51}$ - Minimal pinning sets
Pinning sets for 12^3_51 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_51 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 220
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: 2.98171
on average over minimal pinning sets: 2.48
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 4, 6, 9}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 5, 6, 9, 12}
6
[2, 2, 2, 2, 4, 4]
2.67
b (minimal)
• {1, 2, 5, 6, 8, 9}
6
[2, 2, 2, 2, 4, 4]
2.67
c (minimal)
• {1, 2, 5, 6, 9, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
3
13
2.56
7
0
0
45
2.81
8
0
0
65
2.98
9
0
0
55
3.11
10
0
0
28
3.2
11
0
0
8
3.27
12
0
0
1
3.33
Total
2
3
215
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,6],[0,6,3,3],[0,2,2,7],[0,7,8,8],[1,8,6,1],[1,5,9,2],[3,9,9,4],[4,9,5,4],[6,8,7,7]]
PD code (use to draw this multiloop with SnapPy): [[5,8,6,1],[4,16,5,9],[11,7,12,8],[6,12,7,13],[1,17,2,20],[9,3,10,4],[10,15,11,16],[13,18,14,17],[2,19,3,20],[14,18,15,19]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(11,4,-12,-5)(2,5,-3,-6)(3,12,-4,-13)(10,15,-11,-16)(1,16,-2,-9)(14,19,-15,-20)(7,20,-8,-17)(17,6,-18,-7)(18,13,-19,-14)
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,-9)(-2,-6,17,-8,9)(-3,-13,18,6)(-4,11,15,19,13)(-5,2,16,-11)(-7,-17)(-10,-16,1)(-12,3,5)(-14,-20,7,-18)(-15,10,8,20)(-19,14)(4,12)
Multiloop annotated with half-edges
12^3_51 annotated with half-edges