$12^{3}_{80}$ - Minimal pinning sets
Pinning sets for 12^3_80 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_80 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 144
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97076
on average over minimal pinning sets: 2.31429
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, 7, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 3, 6, 8, 9, 11}
7
[2, 2, 2, 2, 3, 3, 3]
2.43
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.2
6
0
0
7
2.52
7
0
1
21
2.74
8
0
0
39
2.91
9
0
0
41
3.05
10
0
0
25
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
1
1
142
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 7]
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],[0,8,8,1],[1,9,9,2],[3,9,9,7],[3,6,8,8],[4,7,7,4],[5,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[8,14,1,9],[9,15,10,20],[7,19,8,20],[13,18,14,19],[1,16,2,15],[10,6,11,7],[12,4,13,5],[17,3,18,4],[16,3,17,2],[5,11,6,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,9,-1,-10)(10,1,-11,-2)(15,2,-16,-3)(19,4,-20,-5)(3,20,-4,-15)(6,17,-7,-18)(16,11,-17,-12)(5,12,-6,-13)(13,18,-14,-19)(14,7,-9,-8)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,15,-4,19,-14,-8,-10)(-3,-15)(-5,-13,-19)(-6,-18,13)(-7,14,18)(-9,8)(-11,16,2)(-12,5,-20,3,-16)(-17,6,12)(1,9,7,17,11)(4,20)
Multiloop annotated with half-edges
12^3_80 annotated with half-edges