$10^{3}_{6}$ - Minimal pinning sets
Pinning sets for 10^3_6 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^3_6 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 84
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: 2.97851
on average over minimal pinning sets: 2.625
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, 6, 7}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 2, 4, 5, 6, 7}
6
[2, 2, 3, 3, 3, 3]
2.67
b (minimal)
• {1, 4, 5, 6, 7, 9}
6
[2, 2, 3, 3, 3, 3]
2.67
c (minimal)
• {1, 3, 4, 6, 7, 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
0
6
2.73
6
0
3
15
2.85
7
0
0
28
2.98
8
0
0
22
3.09
9
0
0
8
3.17
10
0
0
1
3.2
Total
1
3
80
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: Yes
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,6],[0,7,5,1],[1,4,7,2],[3,7,7,3],[4,6,6,5]]
PD code (use to draw this multiloop with SnapPy): [[4,10,1,5],[5,11,6,16],[3,15,4,16],[9,14,10,15],[1,12,2,11],[6,2,7,3],[13,8,14,9],[12,8,13,7]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(6,13,-7,-14)(14,7,-15,-8)(2,9,-3,-10)(8,15,-9,-16)(1,16,-2,-11)(11,10,-12,-5)(12,3,-13,-4)
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,-11,-5)(-2,-10,11)(-3,12,10)(-4,5,-12)(-6,-14,-8,-16,1)(-7,14)(-9,2,16)(-13,6,4)(-15,8)(3,9,15,7,13)
Multiloop annotated with half-edges
10^3_6 annotated with half-edges