$11^{2}_{4}$ - Minimal pinning sets
Pinning sets for 11^2_4 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_4 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 292
of which optimal: 3
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.06129
on average over minimal pinning sets: 2.725
on average over optimal pinning sets: 2.58333
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {2, 3, 5, 7}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {2, 5, 7, 8}
4
[2, 2, 3, 4]
2.75
C (optimal)
• {2, 5, 6, 9}
4
[2, 2, 3, 3]
2.50
a (minimal)
• {2, 4, 5, 7, 10}
5
[2, 2, 3, 4, 4]
3.00
b (minimal)
• {2, 5, 7, 9, 10}
5
[2, 2, 3, 3, 4]
2.80
c (minimal)
• {2, 4, 5, 6, 7}
5
[2, 2, 3, 3, 4]
2.80
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
3
0
0
2.58
5
0
3
20
2.81
6
0
0
63
2.96
7
0
0
88
3.07
8
0
0
71
3.15
9
0
0
34
3.2
10
0
0
9
3.24
11
0
0
1
3.27
Total
3
3
286
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,4],[0,4,4,5],[0,5,6,7],[0,7,8,5],[0,5,1,1],[1,4,3,2],[2,8,8,7],[2,6,8,3],[3,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[3,8,4,1],[2,18,3,9],[11,7,12,8],[4,16,5,17],[1,10,2,9],[10,17,11,18],[14,6,15,7],[12,15,13,16],[5,13,6,14]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (15,4,-16,-5)(12,5,-13,-6)(1,6,-2,-7)(3,14,-4,-15)(13,16,-14,-17)(2,17,-3,-18)(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,-18,11,7)(-3,-15,-5,12,18)(-4,15)(-6,1,9,-12)(-8,-10)(-9,8,-11)(-13,-17,2,6)(-14,3,17)(-16,13,5)(4,14,16)
Multiloop annotated with half-edges
11^2_4 annotated with half-edges