$10^{1}_{26}$ - Minimal pinning sets
Pinning sets for 10^1_26 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_26 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 124
of which optimal: 2
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98247
on average over minimal pinning sets: 2.65
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)
• {1, 3, 7, 10}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 3, 5, 9}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {1, 3, 6, 7, 9}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {1, 2, 3, 7, 9}
5
[2, 2, 3, 3, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
2
0
0
2.5
5
0
2
12
2.74
6
0
0
34
2.91
7
0
0
40
3.03
8
0
0
25
3.11
9
0
0
8
3.17
10
0
0
1
3.2
Total
2
2
120
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,6,6],[0,6,7,7],[0,7,5,1],[1,4,7,6],[2,5,3,2],[3,5,4,3]]
PD code (use to draw this loop with SnapPy): [[16,9,1,10],[10,6,11,5],[15,4,16,5],[8,13,9,14],[1,7,2,6],[11,2,12,3],[3,14,4,15],[12,7,13,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,16,-10,-1)(13,2,-14,-3)(3,8,-4,-9)(4,15,-5,-16)(10,5,-11,-6)(14,7,-15,-8)(6,11,-7,-12)(1,12,-2,-13)
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,-3,-9)(-2,13)(-4,-16,9)(-5,10,16)(-6,-12,1,-10)(-7,14,2,12)(-8,3,-14)(-11,6)(-15,4,8)(5,15,7,11)
Loop annotated with half-edges
10^1_26 annotated with half-edges