$10^{2}_{15}$ - Minimal pinning sets
Pinning sets for 10^2_15 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_15 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 128
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9872
on average over minimal pinning sets: 2.65
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)
• {2, 3, 7, 8}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 2, 6, 7}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {2, 3, 6, 7, 9}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 2, 4, 7, 8}
5
[2, 2, 3, 3, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.5
5
0
2
12
2.74
6
0
0
35
2.91
7
0
0
42
3.03
8
0
0
26
3.12
9
0
0
8
3.17
10
0
0
1
3.2
Total
2
2
124
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,5,6],[0,6,7,7],[0,7,6,5],[1,4,2,1],[2,4,7,3],[3,6,4,3]]
PD code (use to draw this multiloop with SnapPy): [[10,16,1,11],[11,8,12,7],[9,6,10,7],[3,15,4,16],[1,14,2,13],[8,13,9,12],[2,5,3,6],[14,4,15,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,2,-12,-3)(10,3,-1,-4)(7,4,-8,-5)(5,14,-6,-15)(1,12,-2,-13)(8,13,-9,-14)(15,6,-16,-7)(16,9,-11,-10)
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,-13,8,4)(-2,11,9,13)(-3,10,-11)(-4,7,-16,-10)(-5,-15,-7)(-6,15)(-8,-14,5)(-9,16,6,14)(-12,1,3)(2,12)
Multiloop annotated with half-edges
10^2_15 annotated with half-edges