$11^{2}_{15}$ - Minimal pinning sets
Pinning sets for 11^2_15 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 11^2_15 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 190
of which optimal: 1
of which minimal: 6
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.97507
on average over minimal pinning sets: 2.54167
on average over optimal pinning sets: 2.25
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 2, 8, 10}
4
[2, 2, 2, 3]
2.25
a (minimal)
• {1, 2, 6, 8, 11}
5
[2, 2, 2, 3, 4]
2.60
b (minimal)
• {1, 2, 8, 9, 11}
5
[2, 2, 2, 3, 4]
2.60
c (minimal)
• {1, 2, 5, 8, 11}
5
[2, 2, 2, 3, 4]
2.60
d (minimal)
• {1, 2, 7, 8, 11}
5
[2, 2, 2, 3, 5]
2.80
e (minimal)
• {1, 2, 4, 8, 11}
5
[2, 2, 2, 3, 3]
2.40
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
1
0
0
2.25
5
0
5
7
2.58
6
0
0
36
2.81
7
0
0
55
2.96
8
0
0
50
3.08
9
0
0
27
3.16
10
0
0
8
3.23
11
0
0
1
3.27
Total
1
5
184
Other information about this multiloop Properties
Region degree sequence: [2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,4,2,0],[0,1,5,6],[0,6,4,4],[1,3,3,7],[2,7,8,8],[2,8,7,3],[4,6,8,5],[5,7,6,5]]
PD code (use to draw this multiloop with SnapPy): [[4,18,1,5],[5,3,6,4],[6,17,7,18],[1,14,2,13],[2,12,3,13],[9,16,10,17],[7,15,8,14],[8,11,9,12],[15,10,16,11]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (12,1,-13,-2)(8,17,-9,-18)(15,18,-16,-5)(4,5,-1,-6)(6,3,-7,-4)(7,14,-8,-15)(16,9,-17,-10)(13,10,-14,-11)(2,11,-3,-12)
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 $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12,-3,6)(-2,-12)(-4,-6)(-5,4,-7,-15)(-8,-18,15)(-9,16,18)(-10,13,1,5,-16)(-11,2,-13)(-14,7,3,11)(-17,8,14,10)(9,17)
Multiloop annotated with half-edges
11^2_15 annotated with half-edges