$10^{1}_{17}$ - Minimal pinning sets
Pinning sets for 10^1_17 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_17 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 72
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91204
on average over minimal pinning sets: 2.45
on average over optimal pinning sets: 2.45
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 5, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 3, 4, 5, 9}
5
[2, 2, 2, 3, 4]
2.60
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
4
0
0
2.45
6
0
0
16
2.73
7
0
0
25
2.92
8
0
0
19
3.05
9
0
0
7
3.14
10
0
0
1
3.2
Total
4
0
68
Other information about this loop 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: [[1,1,2,3],[0,4,2,0],[0,1,5,5],[0,6,7,7],[1,7,6,5],[2,4,6,2],[3,5,4,7],[3,6,4,3]]
PD code (use to draw this loop with SnapPy): [[7,16,8,1],[15,6,16,7],[8,6,9,5],[1,13,2,12],[3,14,4,15],[9,4,10,5],[13,10,14,11],[2,11,3,12]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,2,-8,-3)(3,6,-4,-7)(13,4,-14,-5)(11,8,-12,-9)(16,9,-1,-10)(10,15,-11,-16)(1,12,-2,-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,-13,-5,-15,10)(-2,7,-4,13)(-3,-7)(-6,3,-8,11,15)(-9,16,-11)(-10,-16)(-12,1,9)(-14,5)(2,12,8)(4,6,14)
Loop annotated with half-edges
10^1_17 annotated with half-edges