$12^{3}_{83}$ - Minimal pinning sets
Pinning sets for 12^3_83 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_83 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 144
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97193
on average over minimal pinning sets: 2.33333
on average over optimal pinning sets: 2.33333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 3, 4, 5, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
B (optimal)
• {1, 2, 3, 4, 5, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
C (optimal)
• {1, 2, 3, 4, 6, 9}
6
[2, 2, 2, 2, 3, 3]
2.33
D (optimal)
• {1, 2, 3, 4, 6, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
4
0
0
2.33
7
0
0
20
2.66
8
0
0
41
2.89
9
0
0
44
3.06
10
0
0
26
3.18
11
0
0
8
3.27
12
0
0
1
3.33
Total
4
0
140
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,2],[0,1,5,6],[0,7,7,0],[1,7,6,8],[1,9,9,2],[2,8,4,7],[3,6,4,3],[4,6,9,9],[5,8,8,5]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,17,10,20],[7,19,8,20],[15,1,16,2],[17,3,18,4],[10,6,11,7],[13,18,14,19],[2,14,3,15],[4,13,5,12],[5,11,6,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (20,1,-17,-2)(13,4,-14,-5)(10,7,-11,-8)(3,14,-4,-15)(15,2,-16,-3)(8,9,-1,-10)(6,11,-7,-12)(16,17,-9,-18)(5,18,-6,-19)(19,12,-20,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,20,12,-7,10)(-2,15,-4,13,-20)(-3,-15)(-5,-19,-13)(-6,-12,19)(-8,-10)(-9,8,-11,6,18)(-14,3,-16,-18,5)(-17,16,2)(1,9,17)(4,14)(7,11)
Multiloop annotated with half-edges
12^3_83 annotated with half-edges