$11^{2}_{77}$ - Minimal pinning sets
Pinning sets for 11^2_77 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_77 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 168
of which optimal: 6
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.98725
on average over minimal pinning sets: 2.5
on average over optimal pinning sets: 2.5
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 5, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
B (optimal)
• {1, 3, 5, 8, 10}
5
[2, 2, 2, 3, 4]
2.60
C (optimal)
• {1, 2, 5, 8, 10}
5
[2, 2, 2, 3, 3]
2.40
D (optimal)
• {1, 2, 5, 8, 11}
5
[2, 2, 2, 3, 4]
2.60
E (optimal)
• {1, 2, 5, 7, 8}
5
[2, 2, 2, 3, 4]
2.60
F (optimal)
• {1, 2, 4, 5, 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
6
0
0
2.5
6
0
0
27
2.77
7
0
0
50
2.95
8
0
0
49
3.07
9
0
0
27
3.16
10
0
0
8
3.23
11
0
0
1
3.27
Total
6
0
162
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,4,5,2],[0,1,5,6],[0,6,4,0],[1,3,7,7],[1,8,8,2],[2,8,7,3],[4,6,8,4],[5,7,6,5]]
PD code (use to draw this multiloop with SnapPy): [[6,18,1,7],[7,12,8,13],[13,5,14,6],[17,1,18,2],[11,16,12,17],[8,4,9,5],[14,3,15,2],[15,10,16,11],[3,9,4,10]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (7,6,-8,-1)(12,1,-13,-2)(16,3,-17,-4)(4,13,-5,-14)(5,8,-6,-9)(14,9,-15,-10)(18,11,-7,-12)(10,15,-11,-16)(2,17,-3,-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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-7)(-2,-18,-12)(-3,16,-11,18)(-4,-14,-10,-16)(-5,-9,14)(-6,7,11,15,9)(-8,5,13,1)(-13,4,-17,2)(-15,10)(3,17)(6,8)
Multiloop annotated with half-edges
11^2_77 annotated with half-edges