$12^{3}_{85}$ - Minimal pinning sets
Pinning sets for 12^3_85 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_85 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, 8, 9}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 2, 3, 4, 9}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 6, 9, 11}
6
[2, 2, 2, 2, 4, 4]
2.67
b (minimal)
• {1, 2, 3, 6, 9, 12}
6
[2, 2, 2, 2, 4, 4]
2.67
c (minimal)
• {1, 2, 3, 5, 6, 9}
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, 4, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,5],[0,5,6,7],[0,7,4,0],[1,3,8,8],[1,6,2,1],[2,5,9,9],[2,9,8,3],[4,7,9,4],[6,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[6,14,1,7],[7,15,8,20],[9,5,10,6],[13,1,14,2],[15,13,16,12],[8,19,9,20],[4,18,5,19],[10,3,11,2],[16,11,17,12],[17,3,18,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(20,3,-13,-4)(4,17,-5,-18)(12,5,-7,-6)(11,18,-12,-19)(19,10,-20,-11)(2,13,-3,-14)(6,7,-1,-8)(15,8,-16,-9)(9,14,-10,-15)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,8)(-2,-14,9,-16)(-3,20,10,14)(-4,-18,11,-20)(-5,12,18)(-6,-8,15,-10,19,-12)(-7,6)(-9,-15)(-11,-19)(-13,2,-17,4)(1,7,5,17)(3,13)
Multiloop annotated with half-edges
12^3_85 annotated with half-edges