$11^{2}_{66}$ - Minimal pinning sets
Pinning sets for 11^2_66 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_66 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 84
of which optimal: 1
of which minimal: 4
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97572
on average over minimal pinning sets: 2.52857
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)
• {2, 4, 5, 6, 10}
5
[2, 2, 2, 3, 3]
2.40
a (minimal)
• {1, 3, 4, 5, 6, 10, 11}
7
[2, 2, 2, 3, 3, 3, 3]
2.57
b (minimal)
• {1, 3, 4, 5, 6, 8, 11}
7
[2, 2, 2, 3, 3, 3, 3]
2.57
c (minimal)
• {1, 2, 4, 5, 6, 8, 11}
7
[2, 2, 2, 3, 3, 3, 3]
2.57
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.4
6
0
0
6
2.67
7
0
3
15
2.81
8
0
0
28
2.97
9
0
0
22
3.12
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
3
80
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 3, 3, 3, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,5,2],[0,1,5,3],[0,2,6,6],[0,7,5,1],[1,4,7,2],[3,8,8,3],[4,8,8,5],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[4,18,1,5],[5,13,6,12],[3,11,4,12],[17,10,18,11],[1,14,2,13],[6,2,7,3],[9,16,10,17],[14,8,15,7],[15,8,16,9]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (5,4,-6,-1)(6,13,-7,-14)(14,7,-15,-8)(16,9,-17,-10)(1,10,-2,-11)(11,18,-12,-5)(12,3,-13,-4)(8,15,-9,-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,-11,-5)(-2,-18,11)(-3,12,18)(-4,5,-12)(-6,-14,-8,-16,-10,1)(-7,14)(-9,16)(-13,6,4)(-15,8)(-17,2,10)(3,17,9,15,7,13)
Multiloop annotated with half-edges
11^2_66 annotated with half-edges