$10^{2}_{6}$ - Minimal pinning sets
Pinning sets for 10^2_6 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_6 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 98
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: 2.92653
on average over minimal pinning sets: 2.47333
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, 5, 10}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {1, 2, 3, 5, 6}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {2, 3, 5, 8, 9}
5
[2, 2, 2, 3, 3]
2.40
d (minimal)
• {2, 3, 4, 5, 6, 8}
6
[2, 2, 2, 3, 3, 4]
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
3
6
2.62
6
0
1
25
2.81
7
0
0
33
2.97
8
0
0
21
3.07
9
0
0
7
3.14
10
0
0
1
3.2
Total
1
4
93
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,7,7,5],[2,4,6,3],[3,5,7,7],[4,6,6,4]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,7,10,8],[10,15,11,16],[1,6,2,7],[4,14,5,15],[11,5,12,6],[2,12,3,13],[13,3,14,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(16,3,-9,-4)(7,4,-8,-5)(14,5,-15,-6)(8,9,-1,-10)(2,11,-3,-12)(15,12,-16,-13)(6,13,-7,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,10)(-2,-12,15,5,-8,-10)(-3,16,12)(-4,7,13,-16)(-5,14,-7)(-6,-14)(-9,8,4)(-11,2)(-13,6,-15)(1,9,3,11)
Multiloop annotated with half-edges
10^2_6 annotated with half-edges