$11^{2}_{70}$ - Minimal pinning sets
Pinning sets for 11^2_70 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_70 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 312
of which optimal: 4
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.06966
on average over minimal pinning sets: 2.71667
on average over optimal pinning sets: 2.625
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 4, 6, 8}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {2, 4, 8, 10}
4
[2, 2, 3, 4]
2.75
C (optimal)
• {2, 4, 7, 8}
4
[2, 2, 3, 4]
2.75
D (optimal)
• {1, 2, 6, 8}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {2, 4, 8, 9, 11}
5
[2, 2, 3, 3, 4]
2.80
b (minimal)
• {2, 3, 5, 6, 8}
5
[2, 2, 3, 4, 4]
3.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
4
0
0
2.62
5
0
2
24
2.84
6
0
0
68
2.98
7
0
0
94
3.08
8
0
0
75
3.15
9
0
0
35
3.21
10
0
0
9
3.24
11
0
0
1
3.27
Total
4
2
306
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,6,7],[0,7,7,8],[0,8,6,5],[1,4,6,1],[2,5,4,8],[2,8,3,3],[3,7,6,4]]
PD code (use to draw this multiloop with SnapPy): [[4,18,1,5],[5,14,6,15],[15,3,16,4],[10,17,11,18],[1,8,2,7],[13,6,14,7],[2,12,3,13],[16,9,17,10],[11,9,12,8]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(15,18,-16,-5)(9,16,-10,-17)(4,5,-1,-6)(12,7,-13,-8)(17,10,-18,-11)(2,11,-3,-12)(6,13,-7,-14)(14,3,-15,-4)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,8,-13,6)(-2,-12,-8)(-3,14,-7,12)(-4,-6,-14)(-5,4,-15)(-9,-17,-11,2)(-10,17)(-16,9,1,5)(-18,15,3,11)(7,13)(10,16,18)
Multiloop annotated with half-edges
11^2_70 annotated with half-edges