$10^{1}_{6}$ - Minimal pinning sets
Pinning sets for 10^1_6 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_6 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 80
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.89866
on average over minimal pinning sets: 2.325
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 4, 5, 8}
5
[2, 2, 2, 3, 3]
2.40
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
2.25
5
0
1
6
2.54
6
0
0
19
2.76
7
0
0
26
2.93
8
0
0
19
3.05
9
0
0
7
3.14
10
0
0
1
3.2
Total
1
1
78
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,2,2],[0,1,1,3],[0,2,5,6],[0,6,5,1],[3,4,7,7],[3,7,7,4],[5,6,6,5]]
PD code (use to draw this loop with SnapPy): [[16,13,1,14],[14,8,15,7],[15,6,16,7],[12,5,13,6],[1,9,2,8],[2,11,3,12],[4,9,5,10],[10,3,11,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(10,3,-11,-4)(13,4,-14,-5)(5,12,-6,-13)(6,15,-7,-16)(2,7,-3,-8)(16,9,-1,-10)(14,11,-15,-12)
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,-3,10)(-2,-8)(-4,13,-6,-16,-10)(-5,-13)(-7,2,-9,16)(-11,14,4)(-12,5,-14)(-15,6,12)(1,9)(3,7,15,11)
Loop annotated with half-edges
10^1_6 annotated with half-edges