$10^{3}_{5}$ - Minimal pinning sets
Pinning sets for 10^3_5 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^3_5 Pinning data
Pinning number of this multiloop: 6
Total number of pinning sets: 28
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.78265
on average over minimal pinning sets: 2.22222
on average over optimal pinning sets: 2.22222
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, 2, 3]
2.17
B (optimal)
• {1, 2, 3, 5, 8, 9}
6
[2, 2, 2, 2, 2, 4]
2.33
C (optimal)
• {1, 2, 3, 5, 7, 9}
6
[2, 2, 2, 2, 2, 3]
2.17
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
6
3
0
0
2.22
7
0
0
9
2.63
8
0
0
10
2.9
9
0
0
5
3.07
10
0
0
1
3.2
Total
3
0
25
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 6, 6]
Minimal region degree: 2
Is multisimple: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,5,5],[0,6,4,0],[1,3,6,1],[2,7,7,2],[3,7,7,4],[5,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[6,10,1,7],[7,11,8,16],[5,15,6,16],[9,1,10,2],[11,9,12,8],[14,4,15,5],[2,13,3,12],[3,13,4,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(13,4,-14,-5)(16,11,-13,-12)(3,14,-4,-15)(12,5,-7,-6)(6,7,-1,-8)(2,9,-3,-10)(10,15,-11,-16)
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,-10,-16,-12,-6,-8)(-3,-15,10)(-4,13,11,15)(-5,12,-13)(-7,6)(-9,2)(-11,16)(-14,3,9,1,7,5)(4,14)
Multiloop annotated with half-edges
10^3_5 annotated with half-edges