$11^{1}_{101}$ - Minimal pinning sets
Pinning sets for 11^1_101 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_101 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 188
of which optimal: 6
of which minimal: 8
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.0024
on average over minimal pinning sets: 2.65
on average over optimal pinning sets: 2.53333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 4, 6, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 4, 6, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 4, 7, 10}
5
[2, 2, 2, 3, 4]
2.60
D (optimal)
• {1, 4, 7, 8, 10}
5
[2, 2, 2, 4, 4]
2.80
E (optimal)
• {1, 3, 4, 9, 10}
5
[2, 2, 2, 3, 3]
2.40
F (optimal)
• {1, 3, 4, 8, 10}
5
[2, 2, 2, 3, 4]
2.60
a (minimal)
• {1, 4, 6, 8, 10, 11}
6
[2, 2, 2, 3, 4, 5]
3.00
b (minimal)
• {1, 4, 5, 7, 9, 10}
6
[2, 2, 2, 3, 4, 5]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
6
0
0
2.53
6
0
2
30
2.79
7
0
0
59
2.97
8
0
0
54
3.09
9
0
0
28
3.17
10
0
0
8
3.23
11
0
0
1
3.27
Total
6
2
180
Other information about this loop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,7,0],[0,7,8,1],[1,8,8,5],[1,4,6,6],[2,5,5,7],[2,6,8,3],[3,7,4,4]]
PD code (use to draw this loop with SnapPy): [[5,18,6,1],[4,9,5,10],[17,6,18,7],[1,11,2,10],[3,14,4,15],[8,13,9,14],[7,13,8,12],[16,11,17,12],[2,16,3,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (14,3,-15,-4)(1,6,-2,-7)(7,16,-8,-17)(13,8,-14,-9)(9,4,-10,-5)(5,10,-6,-11)(18,11,-1,-12)(12,17,-13,-18)(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)(-17,17)(-18,18)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-7,-17,12)(-2,-16,7)(-3,14,8,16)(-4,9,-14)(-5,-11,18,-13,-9)(-6,1,11)(-8,13,17)(-10,5)(-12,-18)(-15,2,6,10,4)(3,15)
Loop annotated with half-edges
11^1_101 annotated with half-edges