$10^{1}_{5}$ - Minimal pinning sets
Pinning sets for 10^1_5 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 10^1_5 Pinning data
Pinning number of this loop: 4
Total number of pinning sets: 207
of which optimal: 8
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.01465
on average over minimal pinning sets: 2.75
on average over optimal pinning sets: 2.75
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 4, 6}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 3, 5, 8}
4
[2, 2, 4, 4]
3.00
C (optimal)
• {1, 3, 6, 7}
4
[2, 2, 3, 4]
2.75
D (optimal)
• {1, 3, 4, 10}
4
[2, 2, 3, 4]
2.75
E (optimal)
• {1, 3, 7, 10}
4
[2, 2, 4, 4]
3.00
F (optimal)
• {1, 2, 3, 5}
4
[2, 2, 3, 4]
2.75
G (optimal)
• {1, 3, 8, 9}
4
[2, 2, 3, 4]
2.75
H (optimal)
• {1, 2, 3, 9}
4
[2, 2, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
8
0
0
2.75
5
0
0
40
2.9
6
0
0
66
3.0
7
0
0
56
3.07
8
0
0
28
3.12
9
0
0
8
3.17
10
0
0
1
3.2
Total
8
0
199
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: [[1,2,3,3],[0,3,4,5],[0,5,4,3],[0,2,1,0],[1,2,6,7],[1,7,6,2],[4,5,7,7],[4,6,6,5]]
PD code (use to draw this loop with SnapPy): [[16,9,1,10],[10,8,11,7],[15,2,16,3],[8,1,9,2],[11,15,12,14],[6,3,7,4],[12,6,13,5],[13,4,14,5]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,16,-10,-1)(1,8,-2,-9)(3,6,-4,-7)(12,5,-13,-6)(10,7,-11,-8)(4,13,-5,-14)(11,14,-12,-15)(2,15,-3,-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,-9)(-2,-16,9)(-3,-7,10,16)(-4,-14,11,7)(-5,12,14)(-6,3,15,-12)(-8,1,-10)(-11,-15,2,8)(-13,4,6)(5,13)
Loop annotated with half-edges
10^1_5 annotated with half-edges