$11^{2}_{8}$ - Minimal pinning sets
Pinning sets for 11^2_8 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_8 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 80
of which optimal: 1
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90697
on average over minimal pinning sets: 2.26667
on average over optimal pinning sets: 2.2
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 5, 8, 11}
5
[2, 2, 2, 2, 3]
2.20
a (minimal)
• {1, 2, 5, 7, 8, 10}
6
[2, 2, 2, 2, 3, 3]
2.33
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
2.2
6
0
1
6
2.5
7
0
0
19
2.74
8
0
0
26
2.94
9
0
0
19
3.09
10
0
0
7
3.2
11
0
0
1
3.27
Total
1
1
78
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,2,3],[0,3,4,5],[0,6,7,0],[0,4,4,1],[1,3,3,5],[1,4,7,6],[2,5,8,8],[2,8,8,5],[6,7,7,6]]
PD code (use to draw this multiloop with SnapPy): [[8,18,1,9],[9,12,10,13],[17,7,18,8],[1,11,2,12],[10,2,11,3],[13,3,14,4],[4,16,5,17],[6,14,7,15],[15,5,16,6]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(11,2,-12,-3)(13,4,-14,-5)(16,7,-17,-8)(6,17,-7,-18)(15,18,-16,-9)(3,10,-4,-11)(1,12,-2,-13)(5,14,-6,-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,-13,-5,-15,-9)(-2,11,-4,13)(-3,-11)(-6,-18,15)(-7,16,18)(-8,9,-16)(-10,3,-12,1)(-14,5)(-17,6,14,4,10,8)(2,12)(7,17)
Multiloop annotated with half-edges
11^2_8 annotated with half-edges