$10^{2}_{18}$ - Minimal pinning sets
Pinning sets for 10^2_18 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^2_18 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 68
of which optimal: 3
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90854
on average over minimal pinning sets: 2.46667
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 7}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 3, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 4, 6, 7, 9}
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
5
3
0
0
2.4
6
0
1
13
2.69
7
0
0
24
2.9
8
0
0
19
3.05
9
0
0
7
3.14
10
0
0
1
3.2
Total
3
1
64
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,4,2],[0,1,4,5],[0,6,7,0],[1,7,2,1],[2,7,6,6],[3,5,5,7],[3,6,5,4]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,6,10,5],[7,4,8,5],[15,1,16,2],[6,11,7,10],[3,13,4,14],[2,13,3,12],[14,11,15,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,2,-6,-3)(3,14,-4,-15)(12,7,-13,-8)(11,16,-12,-9)(9,8,-10,-1)(1,10,-2,-11)(6,13,-7,-14)(15,4,-16,-5)
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,-9)(-2,5,-16,11)(-3,-15,-5)(-4,15)(-6,-14,3)(-7,12,16,4,14)(-8,9,-12)(-10,1)(-13,6,2,10,8)(7,13)
Multiloop annotated with half-edges
10^2_18 annotated with half-edges