$11^{2}_{11}$ - Minimal pinning sets
Pinning sets for 11^2_11 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_11 Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 124
of which optimal: 5
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90811
on average over minimal pinning sets: 2.32
on average over optimal pinning sets: 2.32
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 4, 5, 10}
5
[2, 2, 2, 2, 3]
2.20
B (optimal)
• {2, 4, 5, 8, 10}
5
[2, 2, 2, 2, 5]
2.60
C (optimal)
• {2, 4, 5, 6, 10}
5
[2, 2, 2, 2, 3]
2.20
D (optimal)
• {2, 4, 5, 7, 10}
5
[2, 2, 2, 2, 4]
2.40
E (optimal)
• {2, 4, 5, 9, 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
5
0
0
2.32
6
0
0
20
2.65
7
0
0
35
2.86
8
0
0
35
3.0
9
0
0
21
3.11
10
0
0
7
3.2
11
0
0
1
3.27
Total
5
0
119
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 7]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,6,7,3],[0,2,8,8],[0,5,1,1],[1,4,6,6],[2,5,5,7],[2,6,8,8],[3,7,7,3]]
PD code (use to draw this multiloop with SnapPy): [[3,8,4,1],[2,18,3,9],[7,15,8,16],[4,15,5,14],[1,10,2,9],[10,17,11,18],[16,11,17,12],[12,6,13,7],[5,13,6,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (13,4,-14,-5)(1,6,-2,-7)(17,12,-18,-13)(3,14,-4,-15)(15,2,-16,-3)(5,16,-6,-17)(11,18,-12,-9)(8,9,-1,-10)(10,7,-11,-8)
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,-7,10)(-2,15,-4,13,-18,11,7)(-3,-15)(-5,-17,-13)(-6,1,9,-12,17)(-8,-10)(-9,8,-11)(-14,3,-16,5)(2,6,16)(4,14)(12,18)
Multiloop annotated with half-edges
11^2_11 annotated with half-edges