$11^{2}_{42}$ - Minimal pinning sets
Pinning sets for 11^2_42 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_42 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 96
of which optimal: 2
of which minimal: 2
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90403
on average over minimal pinning sets: 2.2
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, 3, 5, 6, 10}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {1, 3, 4, 6, 10}
5
[2, 2, 2, 2, 3]
2.20
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
2
0
0
2.2
6
0
0
11
2.55
7
0
0
25
2.79
8
0
0
30
2.97
9
0
0
20
3.1
10
0
0
7
3.2
11
0
0
1
3.27
Total
2
0
94
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,5,5,2],[0,1,3,3],[0,2,2,6],[0,7,8,8],[1,8,6,1],[3,5,7,7],[4,6,6,8],[4,7,5,4]]
PD code (use to draw this multiloop with SnapPy): [[5,8,6,1],[4,18,5,9],[7,17,8,18],[6,17,7,16],[1,14,2,13],[9,3,10,4],[10,15,11,16],[14,11,15,12],[2,12,3,13]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (16,1,-17,-2)(3,6,-4,-7)(10,7,-11,-8)(17,12,-18,-13)(13,4,-14,-5)(5,14,-6,-15)(2,15,-3,-16)(11,18,-12,-9)(8,9,-1,-10)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,16,-3,-7,10)(-2,-16)(-4,13,-18,11,7)(-5,-15,2,-17,-13)(-6,3,15)(-8,-10)(-9,8,-11)(-12,17,1,9)(-14,5)(4,6,14)(12,18)
Multiloop annotated with half-edges
11^2_42 annotated with half-edges