$10^{2}_{16}$ - Minimal pinning sets
Pinning sets for 10^2_16 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_16 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 110
of which optimal: 1
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98695
on average over minimal pinning sets: 2.66667
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)
• {1, 2, 3, 7}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 3, 4, 7, 10}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 3, 4, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 2, 3, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
d (minimal)
• {1, 3, 4, 6, 7, 9}
6
[2, 2, 3, 3, 3, 4]
2.83
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
3
6
2.71
6
0
1
27
2.88
7
0
0
38
3.02
8
0
0
25
3.11
9
0
0
8
3.17
10
0
0
1
3.2
Total
1
4
105
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,4],[0,3,7,5],[1,4,2,1],[2,7,7,3],[3,6,6,4]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,6,10,5],[7,4,8,5],[15,12,16,13],[1,12,2,11],[6,11,7,10],[3,13,4,14],[14,2,15,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(5,2,-6,-3)(3,12,-4,-13)(13,4,-14,-5)(14,7,-15,-8)(10,15,-11,-16)(1,16,-2,-9)(6,11,-7,-12)
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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-9)(-2,5,-14,-8,9)(-3,-13,-5)(-4,13)(-6,-12,3)(-7,14,4,12)(-10,-16,1)(-11,6,2,16)(-15,10,8)(7,11,15)
Multiloop annotated with half-edges
10^2_16 annotated with half-edges