$11^{2}_{71}$ - Minimal pinning sets
Pinning sets for 11^2_71 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_71 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
of which optimal: 3
of which minimal: 3
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.9733
on average over minimal pinning sets: 2.4
on average over optimal pinning sets: 2.4
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 3, 6, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
C (optimal)
• {1, 2, 4, 7, 8}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
3
0
0
2.4
6
0
0
16
2.69
7
0
0
35
2.89
8
0
0
40
3.04
9
0
0
25
3.15
10
0
0
8
3.23
11
0
0
1
3.27
Total
3
0
125
Other information about this multiloop 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],[0,4,5,2],[0,1,5,6],[0,7,7,4],[0,3,8,1],[1,8,8,2],[2,8,7,7],[3,6,6,3],[4,6,5,5]]
PD code (use to draw this multiloop with SnapPy): [[5,12,6,1],[4,18,5,13],[11,17,12,18],[6,15,7,14],[1,14,2,13],[10,3,11,4],[16,8,17,9],[15,8,16,7],[2,9,3,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (8,1,-9,-2)(5,16,-6,-17)(17,6,-18,-7)(12,7,-1,-8)(3,10,-4,-11)(15,4,-16,-5)(9,18,-10,-13)(2,13,-3,-14)(14,11,-15,-12)
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)(-2,-14,-12,-8)(-3,-11,14)(-4,15,11)(-5,-17,-7,12,-15)(-6,17)(-9,-13,2)(-10,3,13)(-16,5)(-18,9,1,7)(4,10,18,6,16)
Multiloop annotated with half-edges
11^2_71 annotated with half-edges