$10^{2}_{8}$ - Minimal pinning sets
Pinning sets for 10^2_8 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_8 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 100
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.978
on average over minimal pinning sets: 2.59167
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, 4, 6}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 5, 8}
5
[2, 2, 3, 3, 3]
2.60
b (minimal)
• {1, 3, 4, 6, 9}
5
[2, 2, 3, 3, 3]
2.60
c (minimal)
• {1, 3, 4, 5, 8, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
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
2
6
2.7
6
0
1
23
2.86
7
0
0
34
3.0
8
0
0
24
3.1
9
0
0
8
3.17
10
0
0
1
3.2
Total
1
3
96
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,2],[0,1,4,0],[0,5,6,1],[1,7,5,2],[3,4,7,6],[3,5,7,7],[4,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[6,16,1,7],[7,14,8,15],[15,5,16,6],[1,13,2,14],[8,4,9,5],[9,12,10,13],[2,10,3,11],[11,3,12,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(15,2,-16,-3)(12,3,-13,-4)(6,7,-1,-8)(16,9,-7,-10)(5,10,-6,-11)(14,11,-15,-12)(4,13,-5,-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)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8)(-2,15,11,-6,-8)(-3,12,-15)(-4,-14,-12)(-5,-11,14)(-7,6,10)(-9,16,2)(-10,5,13,3,-16)(-13,4)(1,7,9)
Multiloop annotated with half-edges
10^2_8 annotated with half-edges