$11^{1}_{55}$ - Minimal pinning sets
Pinning sets for 11^1_55 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^1_55 Pinning data
Pinning number of this loop: 5
Total number of pinning sets: 144
of which optimal: 4
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98138
on average over minimal pinning sets: 2.45
on average over optimal pinning sets: 2.45
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 6, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {2, 4, 5, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 6, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {2, 3, 6, 8, 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
5
4
0
0
2.45
6
0
0
20
2.72
7
0
0
41
2.92
8
0
0
44
3.05
9
0
0
26
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
4
0
140
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: [[1,2,3,4],[0,5,6,2],[0,1,7,3],[0,2,7,7],[0,6,8,8],[1,8,8,6],[1,5,4,7],[2,6,3,3],[4,5,5,4]]
PD code (use to draw this loop with SnapPy): [[18,7,1,8],[8,13,9,14],[14,17,15,18],[15,6,16,7],[1,10,2,11],[3,12,4,13],[9,4,10,5],[5,16,6,17],[2,12,3,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (6,1,-7,-2)(12,3,-13,-4)(15,4,-16,-5)(16,7,-17,-8)(8,17,-9,-18)(18,9,-1,-10)(5,10,-6,-11)(2,13,-3,-14)(11,14,-12,-15)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,6,10)(-2,-14,11,-6)(-3,12,14)(-4,15,-12)(-5,-11,-15)(-7,16,4,-13,2)(-8,-18,-10,5,-16)(-9,18)(-17,8)(1,9,17,7)(3,13)
Loop annotated with half-edges
11^1_55 annotated with half-edges