$10^{1}_{20}$ - Minimal pinning sets
Pinning sets for 10^1_20 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_20 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 104
of which optimal: 1
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.92445
on average over minimal pinning sets: 2.48333
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)
• {2, 4, 5, 9}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {1, 2, 3, 5, 9}
5
[2, 2, 2, 3, 3]
2.40
b (minimal)
• {2, 3, 5, 7, 9}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
• {2, 5, 6, 9, 10}
5
[2, 2, 2, 3, 4]
2.60
d (minimal)
• {2, 3, 5, 6, 9}
5
[2, 2, 2, 3, 3]
2.40
e (minimal)
• {1, 2, 5, 6, 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
2.5
5
0
5
6
2.62
6
0
0
29
2.83
7
0
0
34
2.97
8
0
0
21
3.07
9
0
0
7
3.14
10
0
0
1
3.2
Total
1
5
98
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: [[1,2,2,3],[0,4,5,2],[0,1,6,0],[0,7,4,4],[1,3,3,5],[1,4,7,6],[2,5,7,7],[3,6,6,5]]
PD code (use to draw this loop with SnapPy): [[16,5,1,6],[6,3,7,4],[4,15,5,16],[1,13,2,12],[2,11,3,12],[7,11,8,10],[14,9,15,10],[13,9,14,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (4,1,-5,-2)(14,3,-15,-4)(6,11,-7,-12)(12,7,-13,-8)(8,5,-9,-6)(16,9,-1,-10)(10,15,-11,-16)(2,13,-3,-14)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,4,-15,10)(-2,-14,-4)(-3,14)(-5,8,-13,2)(-6,-12,-8)(-7,12)(-9,16,-11,6)(-10,-16)(1,9,5)(3,13,7,11,15)
Loop annotated with half-edges
10^1_20 annotated with half-edges