$10^{1}_{8}$ - Minimal pinning sets
Pinning sets for 10^1_8 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_8 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, 5}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 3, 5, 9}
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, 3, 3, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,4],[0,5,6,1],[2,7,5,2],[3,4,7,6],[3,5,7,7],[4,6,6,5]]
PD code (use to draw this loop with SnapPy): [[7,16,8,1],[15,6,16,7],[8,6,9,5],[1,14,2,15],[9,4,10,5],[10,13,11,14],[2,11,3,12],[12,3,13,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,1,-11,-2)(7,2,-8,-3)(3,6,-4,-7)(13,4,-14,-5)(16,9,-1,-10)(8,11,-9,-12)(15,12,-16,-13)(5,14,-6,-15)
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)(-2,7,-4,13,-16,-10)(-3,-7)(-5,-15,-13)(-6,3,-8,-12,15)(-9,16,12)(-11,8,2)(-14,5)(1,9,11)(4,6,14)
Loop annotated with half-edges
10^1_8 annotated with half-edges