$11^{1}_{24}$ - Minimal pinning sets
Pinning sets for 11^1_24 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_24 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 184
of which optimal: 5
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.05106
on average over minimal pinning sets: 2.64444
on average over optimal pinning sets: 2.64
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 5, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
B (optimal)
• {1, 2, 5, 8, 9}
5
[2, 2, 3, 3, 3]
2.60
C (optimal)
• {1, 3, 8, 9, 10}
5
[2, 2, 3, 3, 3]
2.60
D (optimal)
• {1, 3, 7, 8, 10}
5
[2, 2, 3, 3, 3]
2.60
E (optimal)
• {1, 3, 6, 8, 9}
5
[2, 2, 3, 3, 4]
2.80
a (minimal)
• {1, 2, 5, 7, 8, 10}
6
[2, 2, 3, 3, 3, 3]
2.67
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
5
0
0
2.64
6
0
1
25
2.85
7
0
0
54
3.0
8
0
0
57
3.11
9
0
0
32
3.19
10
0
0
9
3.24
11
0
0
1
3.27
Total
5
1
178
Other information about this loop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,6],[0,6,7,3],[0,2,7,8],[0,5,5,1],[1,4,4,6],[1,5,8,2],[2,8,8,3],[3,7,7,6]]
PD code (use to draw this loop with SnapPy): [[18,9,1,10],[10,13,11,14],[4,17,5,18],[5,8,6,9],[1,12,2,13],[11,2,12,3],[14,3,15,4],[7,16,8,17],[6,16,7,15]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (11,18,-12,-1)(2,5,-3,-6)(15,6,-16,-7)(16,9,-17,-10)(7,10,-8,-11)(12,3,-13,-4)(4,13,-5,-14)(1,14,-2,-15)(8,17,-9,-18)
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,-15,-7,-11)(-2,-6,15)(-3,12,18,-9,16,6)(-4,-14,1,-12)(-5,2,14)(-8,-18,11)(-10,7,-16)(-13,4)(-17,8,10)(3,5,13)(9,17)
Loop annotated with half-edges
11^1_24 annotated with half-edges