$10^{1}_{3}$ - Minimal pinning sets
Pinning sets for 10^1_3 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_3 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 114
of which optimal: 2
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.91523
on average over minimal pinning sets: 2.525
on average over optimal pinning sets: 2.375
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 7}
4
[2, 2, 2, 3]
2.25
B (optimal)
• {1, 3, 6, 7}
4
[2, 2, 2, 4]
2.50
a (minimal)
• {1, 2, 3, 7, 8}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 3, 7, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
c (minimal)
• {1, 2, 3, 5, 7}
5
[2, 2, 2, 4, 4]
2.80
d (minimal)
• {1, 3, 5, 7, 10}
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
4
2
0
0
2.38
5
0
4
11
2.64
6
0
0
33
2.85
7
0
0
35
2.98
8
0
0
21
3.07
9
0
0
7
3.14
10
0
0
1
3.2
Total
2
4
108
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: [[1,2,3,3],[0,3,4,5],[0,5,6,3],[0,2,1,0],[1,6,6,7],[1,7,7,2],[2,7,4,4],[4,6,5,5]]
PD code (use to draw this loop with SnapPy): [[16,11,1,12],[12,10,13,9],[15,2,16,3],[10,1,11,2],[13,6,14,7],[8,3,9,4],[5,14,6,15],[7,5,8,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,16,-14,-1)(6,1,-7,-2)(2,5,-3,-6)(11,4,-12,-5)(7,10,-8,-11)(14,9,-15,-10)(3,12,-4,-13)(8,15,-9,-16)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,-3,-13)(-2,-6)(-4,11,-8,-16,13)(-5,2,-7,-11)(-9,14,16)(-10,7,1,-14)(-12,3,5)(-15,8,10)(4,12)(9,15)
Loop annotated with half-edges
10^1_3 annotated with half-edges